Inter 2nd Year Maths 2B Integration Important Questions

Students get through Maths 2B Important Questions Inter 2nd Year Maths 2B Integration Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2B Integration Important Questions

Question 1.
\(\int\left(\frac{1}{1-x^{2}}+\frac{2}{1+x^{2}}\right)\) [May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 1

Question 2.
\(\int\) sec2x cosec2x dx on I ⊂ R \ ({nπ : n ∈ Z} ∪ {(2n + 1) \(\frac{\pi}{2}\) : n ∈ Z}) [T.S. Mar. 16; Mar, May 07]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 2

Inter 2nd Year Maths 2B Integration Important Questions

Question 3.
\(\int \frac{1+\cos ^{2} x}{1-\cos 2 x}\) dx on I ⊂ R \ {nπ : n ∈ Z} [Mar. 13]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 3

Question 4.
\(\int \sqrt{1-\cos 2 x}\) dx on I ⊂ [2nπ, (n + 1) π], n ∈ Z [May 06]
Solution:
\(\int \sqrt{1-\cos 2 x}\) dx = \(\int \sqrt{2}\) sin x dx
= –\(\sqrt{2}\) cos x + C

Question 5.
\(\int \frac{1}{\cosh x+\sinh x}\) dx on R. [A.P. Mar. 16]
Solution:
\(\int \frac{1}{\cosh x+\sinh x}\) dx
= \(\int \frac{\cosh x-\sinh x}{\cosh ^{2} x-\sinh ^{2} x}\) dx
= \(\int\) (cosh x – sinh x) dx
= sinh x – cosh x + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 6.
\(\int \frac{1}{1+\cos x}\) dx on I ⊂ R \ {(2n + 1)π : n ∈ Z} [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 4

Question 7.
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx, x ∈ R. [A.P. Mar. 15]
Solution:
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx
t = tan-1 x ⇒ dt = \(\frac{d x}{1+x^{2}}\)
\(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}\)dx = \(\int\) sin t dt
= – cos t + t
= -cos (tan-1 x) + C

Question 8.
\(\int \frac{\log (1+x)}{1+x}\)dx on (-1, ∞) [T.S. Mar. 15]
Solution:
\(\int \frac{\log (1+x)}{1+x}\)dx
t = 1 + x ⇒ dt = dx
Inter 2nd Year Maths 2B Integration Important Questions 5

Question 9.
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\) dx on I = (-1, 1). [May 05]
Solution:
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\)
t = x3 ⇒ dt = mx2 dx
\(\int \frac{x^{2}}{\sqrt{1-x^{6}}}\) = \(\frac{1}{3} \int \frac{d \mathrm{t}}{\sqrt{1-\mathrm{t}^{2}}}\)
= \(\frac{1}{3}\) sin-1 t + C
= \(\frac{1}{3}\) sin-1 (x3) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 10.
\(\int \frac{x^{8}}{1+x^{18}}\) dx on R. [A.P. Mar. 16]
Solution:
t = x9 ⇒ dt = 9x8 dx
\(\int \frac{x^{8} d x}{1+x^{18}}=\int \frac{x^{8}}{1+\left(x^{9}\right)^{2}} d x\)
= \(\frac{1}{9} \int \frac{d t}{1+t^{2}}\) = = \(\frac{1}{9}\) tan-1 t + C
= \(\frac{1}{9}\) tan-1 (x9) + C

Question 11.
\(\int \frac{1}{x \log x[\log (\log x)]}\) dx on (1, ∞) [Mar. 11]
Solution:
t = log (log x)
dt = \(\frac{1}{\log x} \cdot \frac{1}{x}\) dx
\(\int \frac{1}{x \log x[\log (\log x)]}\) dx = \(\int \frac{d t}{t}\)
= log |t| + C
= log |log(log x)| + C

Question 12.
\(\int \frac{1}{(x+3) \sqrt{x+2}}\)dx on I ⊂ (-2, ∞) [Mar. 14]
Solution:
x + 2 = t2
dx = 2t dt
\(\int \frac{d x}{(x+3) \sqrt{x+2}}=\int \frac{2 t d t}{t\left(t^{2}+1\right)}\)
= \(2 \int \frac{d t}{t^{2}+1}\)
= 2 tan-1 (t) + C
= 2 tan-1 (\(\sqrt{x+2}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 13.
\(\int \frac{\cot (\log x)}{x}\) dx, x ∈ I ⊂ (0, ∞) \ {e : n ∈ Z). [Mar. 05]
Solution:
t = log x ⇒ dt = \(\frac{\mathrm{dx}}{\mathrm{x}}\) \(\int\)
\(\int \frac{\cot (\log x)}{x}\) dx = \(\int\) cot t dt = log (sin t) + C
= log |sin (log x)| + C

Question 14.
\(\int\) (tan x + log sec x)ex dx on ((2n – \(\frac{1}{2}\))π, (2n + \(\frac{1}{2}\))π) n ∈ Z. [May 07, Mar. 08]
Solution:
t = log |sec x| ⇒ dt = \(\frac{1}{\sec x}\) . sec x . tan x dx
= tan x dx
\(\int\) (tan x + log sec x)ex dx = ex . log|sec x| + C

Question 15.
\(\int \sqrt{x}\) log x dx on (0, ∞) [T.S. Mar. 16]
Solution:
\(\int \sqrt{x}\) log x dx
= (log x) \(\frac{2}{3}\) x3/2 – \(\frac{2}{3}\) \(\int\) x3/2 \(\frac{1}{x}\) dx
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{2}{3}\) \(\int\) x1/2 dx
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{2}{3}\) \(\frac{x^{3 / 2}}{3 / 2}\) + C
= \(\frac{2}{3}\) x3/2 (log x) – \(\frac{4}{9}\) x3/2 + C

Question 16.
\(\int\) ex (tan x + sec2 x)dx on I ⊂ R \ {(2n + 1)\(\frac{\pi}{2}\) : n ∈ Z} [Mar 06]
Solution:
f(x) = tanx = f'(x) ⇒ sec2 x dx
I = \(\int\) ex [f(x) + f'(x)] dx = ex. f(x) + C
= ex . tan x + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 17.
\(\int e^{x}\left(\frac{1+x \log x}{x}\right)\) dx on (0, ∞) [A.P. Mar. 15, Mar. 13]
Solution:
\(\int e^{x}\left(\frac{1+x \log x}{x}\right)\) dx = \(\int\) ex (log x + \(\frac{1}{x}\))dx
= ex . log x + C

Question 18.
\(\int \frac{\cos x}{\sin ^{2} x+4 \sin x+5}\)dx [Mar. 07]
Solution:
t = sin x ⇒ dt = cos x dx
I = \(\int \frac{d t}{t^{2}+4 t+5}=\int \frac{d t}{(t+2)^{2}+1}\)
= tan-1(t + 2) + C
= tan-1(sin x + 2) + C

Question 19.
\(\int \frac{d x}{(x+1)(x+2)}\) [Mar. 14, May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 6

Question 20.
\(\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) dx on I ⊂ R \ {x ∈ R : cos (xex) = 0} [T.S. Mar. 17]
Solution:
t = x . ex
dt = (x . ex + ex) dx = ex (1 + x) dx
\(\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) dx = \(\int \frac{d t}{\cos ^{2} t}=\int \sec ^{2} t d t\)
= tan t + C
= tan (x . ex) + C

Question 21.
\(\int\) x tan-1 x dx, x∈ R [Mar. 05]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 7

Inter 2nd Year Maths 2B Integration Important Questions

Question 22.
\(\int \sqrt{1+3 x-x^{2}} d x\) [May 11]
Solution:
\(\int \sqrt{1+3 x-x^{2}} d x=\int \sqrt{1-\left(x^{2}-3 x\right)} d x\)
= \(\int \sqrt{1-\left(x^{2}-3 x+\frac{9}{4}-\frac{9}{4}\right)} d x\)
Inter 2nd Year Maths 2B Integration Important Questions 8

Question 23.
\(\int \frac{9 \cos x-\sin x}{4 \sin x+5 \cos x} d x\) [T.S. Mar. 17; Mar. 08]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 9

Question 24.
\(\int \frac{d x}{5+4 \cos 2 x}\) [Mar. 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 10

Question 25.
Obtain reduction formula for In = \(\int\) cotn x dx, n being a positive integer. n ≥ 2 and deduce the value of \(\int\) cot4 x dx. [T.S. Mar. 19] [A.P. Mar. 16; May 11]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 11

Inter 2nd Year Maths 2B Integration Important Questions

Question 26.
Obtain reduction formula for In = \(\int\) cosecn x dx, n being a positive integer. n ≥ 2 and deduce the value of \(\int\) cosec5 x dx. [T.S. Mar. 16]
Solution:
In = \(\int\) cosecn x dx = \(\int\) cosecn – 2x . cosec2 x dx
= cosecn – 2x (-cot x) + \(\int\) cot x . (n – 2) cosecn – 3 . (cot x) dx
= – cosecn – 2x . cot x + (n – 2) \(\int\) cosecn – 2 x . (cosec2 x -1) dx
= – cosecn – 2x . cot x + (n – 2) In – 2 – (n – 2)In
In (1 + n – 2) = – cosecn – 2 x . cot x + (n – 2) In – 2
Inter 2nd Year Maths 2B Integration Important Questions 12

Question 27.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\) dx. [T.S. Mar. 15]
Solution:
We write
2x + 5 = A \(\frac{\mathrm{d}}{\mathrm{dx}}\) (x2 – 2x + 10) + B
= A(2x – 2) + B
On comparing the coefficients of the like powers of x on both sides of the above equation, we get A = 1 and B = 7.
Thus 2x + 5 = (2x – 2) + 7
Hence \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\) dx
= \(\int \frac{2 x-2}{\sqrt{x^{2}-2 x+10}} d x+7 \int \frac{d x}{\sqrt{x^{2}-2 x+10}}+C\)
= \(2 \sqrt{x^{2}-2 x+10}+7 \int \frac{d x}{\sqrt{(x-1)^{2}+3^{2}}}\) + C
= \(2 \sqrt{x^{2}-2 x+10}+7 \sinh ^{-1}\left(\frac{x-1}{3}\right)\) + C

Question 28.
Evaluate \(\int\) sin4 x dx. [Mar. 14]
Solution:
In = sinn x dx = \(-\frac{\sin ^{n-1} x \cdot \cos x}{n}+\frac{n-1}{n} \cdot I_{n-2}\)
Inter 2nd Year Maths 2B Integration Important Questions 13

Inter 2nd Year Maths 2B Integration Important Questions

Question 29.
\(\int \frac{2 \cos x+3 \sin x}{4 \cos x+5 \sin x} d x\) [A.P. Mar. 15]
Solution:
Let 2 cos x + 3 sin x = A(4 cos x + 5 sin x) + B(-4 sin x + 5 cos x)
Equating the coefficient of sin x and cos x we get
4A + 5B = 2
5A – 4B = 3
Inter 2nd Year Maths 2B Integration Important Questions 14

Question 30
\(\int \frac{1}{1+\sin x+\cos x}\) dx [T.S. Mar. 15]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 15

Question 31.
\(\int\) (6x + 5) \(\sqrt{6-2 x^{2}+x}\) dx [May 06]
Solution:
Let 6x + 5 = A(1 – 4x) + B
Equating the co-efficients of x
6 = -4 A ⇒ A = \(\frac{-3}{2}\)
Equating the constants
A + B = 5
B = 5 – A = 5 + \(\frac{3}{2}\) = \(\frac{13}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 16
Inter 2nd Year Maths 2B Integration Important Questions 17

Inter 2nd Year Maths 2B Integration Important Questions

Question 32.
\(\int \frac{d x}{4+5 \sin x}\) [Mar. 05]
Solution:
t = tan \(\frac{x}{2}\) ⇒ dt = sec2 \(\frac{x}{2}\) . \(\frac{1}{2}\) dx
Inter 2nd Year Maths 2B Integration Important Questions 18
Inter 2nd Year Maths 2B Integration Important Questions 19

Question 33.
\(\int \frac{d x}{(1+x) \sqrt{3+2 x-x^{2}}}\) [May 05]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 20
Inter 2nd Year Maths 2B Integration Important Questions 21

Question 34.
\(\int \frac{d x}{4 \cos x+3 \sin x}\) [Mar. 06]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 22
Inter 2nd Year Maths 2B Integration Important Questions 23

Inter 2nd Year Maths 2B Integration Important Questions

Question 35.
\(\int \frac{2 \sin x+3 \cos x+4}{3 \sin x+4 \cos x+5} d x\) [Mar. 14, 11] [A.P. & T.S. Mar. 16]
Solution:
Let 2 sin x + 3 cos x + 4
= A(3 sin x+4 cos x + 5) + 3(3 cos x – 4sin x) + C
Equating the co-efficients of
sin x. we get 3A – 4B = 2
cos x, we get 4A + 3B = 3
Inter 2nd Year Maths 2B Integration Important Questions 24
Inter 2nd Year Maths 2B Integration Important Questions 25
Substituting in (1)
I = \(\frac{18}{25}\) . x + \(\frac{1}{25}\) log |3 sin x + 4 cos x + 5| – \(\frac{4}{5\left(3+\tan \frac{x}{2}\right)}\) + C

Question 36.
\(\int \frac{x+3}{(x-1)\left(x^{2}+1\right)} d x\) dx [May 07]
Solution:
Let \(\frac{x+3}{(x-1)\left(x^{2}+1\right)}\) = \(\frac{A}{x-1}\) + \(\frac{B x+C}{x^{2}+1}\)
⇒ (x + 3) = A(x2 + 1) + (Bx + C)(x – 1) ………………… (1)
Put x = 1 in (1)
Then 4 = A(1 + 1) + 0 ⇒ A = 2
Put x = 0 in (1)
3 = A(1) + C(-1)
⇒ A – C = 3 ⇒ C = A – 3 = 2 – 3 = -1
Equating coefficient of x2 in (1)
0 = A + B
⇒ B = -A = -2
Inter 2nd Year Maths 2B Integration Important Questions 26

Inter 2nd Year Maths 2B Integration Important Questions

Question 37.
Find \(\int \frac{d x}{3 \cos x+4 \sin x+6}\) [Mar. 13]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 27
Inter 2nd Year Maths 2B Integration Important Questions 28

Question 38.
Find \(\int\) 2x7 dx on R.
Solution:
\(\int\) 2x7 dx = 2 \(\int\) x7 dx
= 2 . \(\frac{x^{8}}{8}\) + C
= \(\frac{x^{8}}{4}\) + C

Question 39.
Evaluate \(\int\) cot2x dx on I ⊂ R \ {nπ : n ∈ Z}.
Solution:
\(\int\) cot2x dx = \(\int\) (cosec2x – 1) dx
= \(\int\) cosec2 x dx – \(\int\) dx
= -cot x – x + C

Question 40.
Evaluate \(\int\left(\frac{x^{6}-1}{1+x^{2}}\right)\) dx for x ∈ R.
Solution:
\(\int\left(\frac{x^{6}-1}{1+x^{2}}\right)\) dx = \(\int\)[(x4 – x2 + 1) – \(\frac{2}{1+x^{2}}\)] dx
= \(\int\) x4 dx – \(\int\) x2 dx + \(\int\) dx – 2 \(\int \frac{d x}{1+x^{2}}\)
= \(\frac{x^{5}}{5}\) – \(\frac{x^{3}}{3}\) + x – 2 tan-1 x + C.

Inter 2nd Year Maths 2B Integration Important Questions

Question 41.
Find \(\int\) (1 – x) (4 – 3x) (3 + 2x) dx, x ∈ R.
Solution:
(1 – x) (4 – 3x) (3 + 2x) = 6x3 – 5x2 – 13x + 12
\(\int\)(1 – x) (4 – 3x) (3 + 2x) dx
= \(\int\) (6x3 – 5x2 – 13x + 12) dx
= 6\(\int\) x3dx – 5 \(\int\) x2 dx – 13 \(\int\) x dx + 12 \(\int\) dx
= \(\frac{6 x^{4}}{4}\) – 5 \(\frac{x^{3}}{3}\) – \(\frac{13 x^{2}}{2}\) + 12x + C
= \(\frac{3}{2}\)x4 – \(\frac{5}{3}\)x3 – \(\frac{13}{2}\)x2 + 12x + C.

Question 42.
Evaluate \(\int\left(x+\frac{1}{x}\right)^{3}\) dx, x > 0.
Solution:
(x + \(\frac{1}{x}\))3 = x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^{3}}\)
\(\int\) (x + + \(\frac{1}{x}\))3 dx = \(\int\) (x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^{3}}\)) dx
= \(\int\) x3 dx + 3 \(\int\) x dx + 3 \(\int\) \(\frac{\mathrm{dx}}{\mathrm{x}}\) + \(\int\) \(\frac{d x}{x^{3}}\)
= \(\frac{x^{4}}{4}\) + \(\frac{3 x^{2}}{2}\) + 3 log x – \(\frac{1}{2 x^{2}}\) + C

Question 43.
Find \(\int \sqrt{1+\sin 2 x}\) dx on R.
Solution:
1 + sin 2x = sin2 x + cos2 x + 2 sin x . cos x
= (sin x + cos x)2
\(\sqrt{1+\sin 2 x}\) = sin x + cos x
If 2nπ – \(\frac{\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{3\pi}{4}\)
= -(sin x + cos x). otherwise
If 2nπ – \(\frac{\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{3\pi}{4}\), then
\(\int \sqrt{1+\sin 2 x}\) dx = \(\int\) (sin x + cos x)dx
= -cos x + sin x + C
If 2nπ + \(\frac{3\pi}{4}\) ≤ x ≤ 2nπ + \(\frac{7\pi}{4}\)
\(\int \sqrt{1+\sin 2 x}\)
= \(\int\) -(sin x + cos x)dx
= –\(\int\) sin x dx – \(\int\) cos x dx
= cos x – sin x + c

Question 44.
Evaluate \(\int \frac{2 x^{3}-3 x+5}{2 x^{2}}\) dx for x > 0 and verify the result by differentiation.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 29
and it is the given expression and hence the result is correct.

Inter 2nd Year Maths 2B Integration Important Questions

Question 45.
Evaluate \(\int \frac{x^{5}}{1+x^{12}}\) dx on R.
Solution:
We define f : R → R by f(t) = \(\frac{1}{1+t^{2}}\)
g : R → R by g(x) = x6
Then g'(x) = 6x5
Define F : R → R by F(t) = tan-1 t
F is the primitive of f
\(\int \frac{x^{5}}{1+x^{12}}\) dx = \(\frac{1}{6}\) \(\int\) f(g(x)) g'(x) dx
= \(\frac{1}{6}\) (F(t) + C)t=g(x)
= \(\frac{1}{6}\) [tan-1 t + C]t=x6
= \(\frac{1}{6}\) tan-1 x6 + C

Question 46.
Evaluate \(\int\) cos3 x sin x dx on R.
Solution:
We define : f = R → R by f(x) = cosx
∴ f'(x) = – sin x
\(\int\) cos3 x sin x dx = \(\int\) (f(x))3 [-f'(x)] dx
= \(\frac{-[f(x)]^{4}}{4}\) + C
= \(\frac{-\cos ^{4} x}{4}\) + C

Question 47.
Find \(\int\) (1 – \(\frac{1}{x^{2}}\)) e(x + \(\frac{1}{x}\)) dx on I where I = (0, ∞)
Solution:
Let J = I = (0, ∞)
Define f : I → R by f(t) = et and g : J → R by g(x) = x + \(\frac{1}{x}\)
Then g(J) ⊂ I, g'(x) = 1 – \(\frac{1}{x^{2}}\)
\(\int\) (1 – \(\frac{1}{x^{2}}\)) e(x + \(\frac{1}{x}\)) dx = \(\int\) f (g(x)) g'(x) dx
= \(\int\) [f(t)dt]t = g(x)
= [\(\int\)et dt]t = g(x)
= [et + c]t=x+\(\frac{1}{x}\)
= e(x+\(\frac{1}{x}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 48.
Evaluate \(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^{2}}}\) dx on I = (0, 1).
Solution:
We define f : I → R by f(x) = sin-1x
f'(x) = \(\frac{1}{\sqrt{1-x^{2}}}\)
\(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^{2}}}\) dx = \(\int \frac{f^{\prime}(x)}{\sqrt{f(x)}} d x\)
= 2\(\sqrt{f(x)}\) + C
= 2 \(\sqrt{\sin ^{-1} x}\) + C

Question 49.
Evaluate \(\int \frac{\sin ^{4} x}{\cos ^{6} x}\) dx, x ∈ I ⊂ R \ {\(\frac{(2 n+1) \pi}{2}\) : n ∈ z}
Solution:
\(\int \frac{\sin ^{4} \dot{x}}{\cos ^{6} x}\) dx = \(\int\) tan4 x . sec2 x dx
We define f : I → R by f(x)
f(x) = tan x, then f'(x) = sec2x
\(\int \frac{\sin ^{4} \dot{x}}{\cos ^{6} x}\) dx = \(\int\) [f(x)]4 . f'(x) dx = \(\frac{[f(x)]^{5}}{5}\) + C.
= \(\frac{1}{5}\) tan5x + C.

Question 50.
Evaluate \(\int\) sin2 x dx on R.
Solution:
\(\int\) sin2x dx = \(\int \frac{(1-\cos 2 x)}{2}\) dx
= \(\frac{1}{2}\) \(\int\) dx – \(\frac{1}{2}\) \(\int\) cos 2x dx
= \(\frac{1}{2}\) x – \(\frac{1}{4}\) sin 2x + C.
(since \(\int\) cos 2x dx = \(\frac{1}{2}\) sin 2x + C)

Question 51.
Evaluate \(\int \frac{1}{a \sin x+b \cos x}\) dx where a, b ∈ R and a2 + b2 ≠ 0 on R.
Solution:
We can find real numbers r and θ such that
a = r cos θ, b = r sin θ
Then r = \(\sqrt{a^{2}+b^{2}}\), cos θ = \(\frac{a}{r}\) and sin θ = \(\frac{b}{r}\)
a sin x + b cos x = r . cos θ sin x + r sin θ cos x
= r[cos θ sin x + sin θ cos x]
= r sin (x + θ)
\(
= [latex]\frac{1}{r}\) (cosec (x + θ) dx
= \(\frac{1}{r}\) log |tan \(\frac{1}{2}\)(x + θ)| + C
= \(\frac{1}{\sqrt{a^{2}+b^{2}}} \log \left|\tan \frac{1}{2}(\tilde{x}+\theta)\right|+c\)
For all x ∈ I where I is an interval disjoint with {nπ – θ : n ∈ z}.

Inter 2nd Year Maths 2B Integration Important Questions

Question 52.
Find \(\int \frac{x^{2}}{\sqrt{x+5}}\) dx on (-5, ∞)
Solution:
Put t = x + 5 so that t > 0 on (-5, ∞)
dx = dt and x = t – 5
Inter 2nd Year Maths 2B Integration Important Questions 30

Question 53.
Find \(\int \frac{x}{\sqrt{1-x}}\) dx, x ∈ I = (0, 1)
Solution:
We define f : I → R by f(x) = \(\frac{x}{\sqrt{1-x}}\)
Let J = (0, \(\frac{\pi}{2}\))
Define Φ : J → I by Φ(θ) = sin2 θ
Then Φ is a bijective mapping from J to I Further Φ and Φ-1 are differentiable on their respective domains.
put x = Φ(θ) = sin2θ
dx = 2 sin θ . cos θ dθ
Inter 2nd Year Maths 2B Integration Important Questions 31

Question 54.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) on (-4, ∞)
Solution:
Let I = (-4, ∞)
Define f on I as f(x) = \(\frac{d x}{(x+5) \sqrt{x+4}}\)
Let J = (0, ∞)
We define Φ : J → I by Φ(t) = t2 – 4
Then Φ is differentiable and is a bijection
Put x = Φ(t) = t2 – 4
then t = \(\sqrt{x+4}\) ⇒ dx = 2t dt
Thus \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) = \(\int \frac{2 t d t}{\left(t^{2}+1\right) t}\)
= \(\int \frac{2 d t}{t^{2}+1}\)
= 2 tan-1 t + C
= 2 tan-1 (\(\sqrt{x+4}\)) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 55.
Evaluate \(\int \frac{d x}{\sqrt{4-9 x^{2}}}\) on I = (-\(\frac{2}{3}\), \(\frac{2}{3}\))
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 32

Question 56.
Evaluate \(\int \frac{1}{a^{2}-x^{2}}\) dx for x ∈ I = (-a, a)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 33

Question 57.
Evaluate \(\int \frac{1}{1+4 x^{2}}\) dx on R.
Solution:
\(\int \frac{1}{1+4 x^{2}}\) dx = \(\int \frac{d x}{4\left(\left(\frac{1}{2}\right)^{2}+x^{2}\right)}\)
= \(\frac{1}{4}\) (2 tan-1 2x) + C
= \(\frac{1}{2}\) tan-1 2x + C

Question 58.
Find \(\int \frac{1}{\sqrt{4-x^{2}}}\) dx on (-2, 2).
Solution:
\(\int \frac{1}{\sqrt{4-x^{2}}}\) dx = \(\int \frac{1}{\sqrt{2^{2}-x^{2}}}\) dx = sin-1(\(\frac{x}{2}\) dx) + C

Question 59.
Evaluate \(\int \sqrt{4 x^{2}+9}\) dx on R.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 34

Inter 2nd Year Maths 2B Integration Important Questions

Question 60.
Evaluate \(\int \sqrt{9 x^{2}-25} d x\) on [\(\frac{5}{3}\), ∞)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 35

Question 61.
Evaluate \(\int \sqrt{16-25 x^{2}}\) dx on (\(\frac{-4}{5}\), \(\frac{4}{5}\))
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 36

Question 62.
Evaluate \(\int\) x sin-1x dx on (-1, 1).
Solution:
Let u(x) = sin-1 x and v(x) = \(\frac{x^{2}}{2}\) so that
v'(x) = x
∴ u(x) v'(x) = x sin-1x
Even though the domain of u is (-1, 1) the function u ¡s differentiable only on (-1, 1).
From the same formula, we have
Inter 2nd Year Maths 2B Integration Important Questions 37
Inter 2nd Year Maths 2B Integration Important Questions 38

Inter 2nd Year Maths 2B Integration Important Questions

Question 63.
Evaluate \(\int\) x2 cos x dx.
Solution:
Let us take u(x) = x2, v(x) = sin x
so that v'(x) = cos x
u(x) v'(x) = x2 cosx
By using the formula for integration by parts, we have
\(\int\) x2 cos x dx = x2 sin x – \(\int\) sin x (x2)’ dx
= x2 sin x – 2 \(\int\) x sin x dx + C.
Again, by applying the formula for integration by parts to
\(\int\) x sin x dx, we get
\(\int\) x. sin x dx = -x cos x – \(\int\) (-cos x) dx
= -x cos x + sin x + C2
\(\int\) x2 cos x dx = x2 sin x – 2(sin x – x cos x) + C
= x2 sin x – 2 sin x + 2x cosx + C
= (x2 – 2) sin x + 2x cos x + C
In evaluating certain integrals by using the formula for integration by parts,, twice or more than twice, we come across the given integral with change of sign. This enables us to evaluate the given integral.

Question 64.
Evaluate \(\int\) ex sin x dx on R.
Solution:
Let A = ex sin x dx on R
A = \(\int\) ex . sin x dx = \(\int\) ex (-cos x)’ dx.
= ex (-cos x) – \(\int\) (-cos x) (ex)’ dx
= – ex cos x + \(\int\) ex cos x dx + C1 ………(1)
\(\int\) ex cos x dx = ex. sin x – \(\int\) ex . sin x dx
= ex sinx – A …………….. (2)
From (1) and (2)
A = – ex cos x + ex sin x – A + C1
2A = ex (sin x – cos x) + C1
A = \(\frac{1}{2}\) ex (sin x – cos x) + C where
C = \(\frac{C_{1}}{2}\)
i.e., \(\int\) ex sin x dx = \(\frac{1}{2}\) ex (sin x – cos x) + C.

Inter 2nd Year Maths 2B Integration Important Questions

Question 65.
Find \(\int\) eax cos (bx + c) dx on R, where a. b, c are real numbers and b ≠ 0.
Solution:
Let A = \(\int\) eax cos (bx + c)dx
Then from the formula for integration by parts
A = eax [latex]\frac{\sin (b x+c)}{b}[/latex] – \(\int\) a eax [latex]\frac{\sin (b x+c)}{b}[/latex] dx
= \(\frac{1}{b}\) eax sin(bx + c) – \(\frac{a}{b}\) \(\int\) eax . sin(bx + c) dx
Inter 2nd Year Maths 2B Integration Important Questions 39
By taking c = 0, we get
\(\int\) eax . cos bx dx = \(\frac{e^{a x}}{a^{2}+b^{2}}\) [a cos bx + b sin bx] + K

Question 66.
Evaluate \(\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\) dx, on (-1, 1)
Solution:
Put x = cos θ, θ ∈ (0, π) dx = -sin θ . dθ
Inter 2nd Year Maths 2B Integration Important Questions 40

Question 67.
Evaluate \(\int e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x\) on I ⊂ R \ {2nπ : n ∈ z}.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 41
= -ex . cot \(\frac{x}{2}\) + C

Inter 2nd Year Maths 2B Integration Important Questions

Question 68.
Evaluate \(\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x\) on I ⊂ R \ {-1, 1}
Solution:
Let x = tan θ ⇒ dx = sec2 θ dθ
\(\frac{2 x}{1-x^{2}}=\frac{2 \tan \theta}{1-\tan ^{2} \theta}\) = tan 2θ
tan-1 \(\left(\frac{2 x}{1-x^{2}}\right)\) = tan-1 (tan 2θ ) = 2θ + nπ
Where n = 0 if |x| < 1 = -1 if x > 1
= 1 if x < -1
We have dθ = \(\frac{1}{1+x^{2}}\) dx and
1 + x2 = 1 + tan2 θ = sec2θ
∴ \(\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x\)
= \(\int\left(\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right)\left(1+x^{2}\right) \frac{1}{1+x^{2}} d x\)
= \(\int\) (2θ + nπ) \(\int\) sec2θ dθ
= 2 \(\int\) θ sec2 θ dθ + nπ ) \(\int\) sec2 θ dθ + c
= 2 (θ tan θ – \(\int\) tan θ dθ) nπ tan θ + c
= 2 (θ tan θ + log |cos θ| + nπ tan θ + c
= (2θ + nπ) tan θ + 2 log cos θ + c
= (2θ + nπ) tan θ + log cos2 θ + c
= (2θ + nπ) tan θ + log sec2 θ + c
= x tan-1 \(\left(\frac{2 x}{1-x^{2}}\right)\) – log (1 + x2) + c

Question 69.
Find \(\int x^{2} \cdot \frac{\exp \left(m \sin ^{-1} x\right)}{\sqrt{1-x^{2}}}\)dx on (-1, 1) where m is a real number. (Here for y ∈ R, exp. {y} stands for ey).
Solution:
Let t = sin-1x, then
Inter 2nd Year Maths 2B Integration Important Questions 42
Inter 2nd Year Maths 2B Integration Important Questions 43
Inter 2nd Year Maths 2B Integration Important Questions 44

Inter 2nd Year Maths 2B Integration Important Questions

Question 70.
Evaluate \(\int \frac{d x}{4 x^{2}-4 x-7}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 45

Question 71.
Find \(\int \frac{d x}{5-2 x^{2}+4 x}\)
Solution:
5 – 2x2 + 4x = -2 (x2 – 2x – \(\frac{5}{2}\))
= -2 ((x – 1)2 – \(\frac{5}{2}\) – 1)
= -2 ((x – 1)2 – \(\left(\sqrt{\frac{7}{2}}\right)^{2}\))
\(\int \frac{d x}{5-2 x^{2}+4 x}\)
= \(-\frac{1}{2} \int \frac{1}{\left((x-1)^{2}-\sqrt{\frac{7}{2}}\right)^{2}} d x+C\)
Inter 2nd Year Maths 2B Integration Important Questions 46

Question 72.
Evaluate \(\int \frac{d x}{x^{2}+x+1}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 47

Question 73.
Evaluate \(\int \frac{d x}{\sqrt{x^{2}+2 x+10}}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 48

Inter 2nd Year Maths 2B Integration Important Questions

Question 74.
Evaluate \(\int \frac{d x}{\sqrt{1+x-x^{2}}}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 49

Question 75.
Evaluate \(\int \sqrt{3+8 x-3 x^{2}} d x\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 50
Inter 2nd Year Maths 2B Integration Important Questions 51
Inter 2nd Year Maths 2B Integration Important Questions 52

Question 76.
Evaluate \(\int \frac{x+1}{x^{2}+3 x+12}\) dx.
Solution:
We write x + 1 = A(2x + 3) + B
Equating the co-efficients of x; we get 1 = 2A.
A = \(\frac{1}{2}\)
Equating the constants 3A + B = 1
B = 1 – 3A = 1 – \(\frac{3}{2}\) = –\(\frac{1}{2}\)
x + 1 = \(\frac{1}{2}\) (2x + 3) – \(\frac{1}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 53

Inter 2nd Year Maths 2B Integration Important Questions

Question 77.
Evaluate \(\int(3 x-2) \sqrt{2 x^{2}-x+1} d x\)
Solution:
Let (3x – 2) = A(4x – 1) + B
Equating the co-efficients of x, we get 3 = 4A
A = \(\frac{3}{4}\)
Equating the constants -2 = -A + B
B = -2 + A = -2 + \(\frac{3}{4}\)
= \(\frac{-5}{4}\)
Inter 2nd Year Maths 2B Integration Important Questions 54
Inter 2nd Year Maths 2B Integration Important Questions 55

Question 78.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\)dx. [T.S. Mar. 15]
Solution:
We write
2x + 5 = A \(\frac{\mathrm{d}}{\mathrm{dx}}\) (x2 – 2x + 10) + B
= A (2x – 2) + B
On comparing the coefficients of the like powers of x on both sides of the above equation, we get A = 1 and B = 7.
Thus 2x + 5 = (2x – 2) + 7
Hence \(\int \frac{2 x+5}{\sqrt{x^{2}-2 x+10}}\)dx
= \(\int \frac{2 x-2}{\sqrt{x^{2}-2 x+10}} d x+7 \int \frac{d x}{\sqrt{x^{2}-2 x+10}}+C\)
Inter 2nd Year Maths 2B Integration Important Questions 56

Inter 2nd Year Maths 2B Integration Important Questions

Question 79.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\)
Solution:
Put t = \(\sqrt{x+4}\)
dt = \(\frac{1}{2 \sqrt{x+4}}\) dx
We have t2 = x + 4
x + 5 = t2 + 1
\(\int \frac{d x}{(x+5)(\sqrt{x+4})}=\int \frac{2}{t^{2}+1} d t\)
= 2 tan-1 t + C
= 2 tan-1 (\(\sqrt{x+4}\)) + C.

Question 80.
Evaluate \(\int \frac{d x}{5+4 \cos x}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 57

Question 81.
Find \(\int \frac{d x}{3 \cos x+4 \sin x+6}\) [Mar. 13]
Solution:
t = tan \(\frac{x}{2}\) ⇒ dx = \(\frac{2 d t}{1+t^{2}}\)
Inter 2nd Year Maths 2B Integration Important Questions 58
Inter 2nd Year Maths 2B Integration Important Questions 59

Question 82.
Find \(\int \frac{d x}{d+e \tan x}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 60
Inter 2nd Year Maths 2B Integration Important Questions 61

Inter 2nd Year Maths 2B Integration Important Questions

Question 83.
Evaluate \(\int \frac{\sin x}{d \cos x+e \sin x} d x\) and \(\int \frac{\cos x}{d \cos x+e \sin x} d x\).
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 62
Inter 2nd Year Maths 2B Integration Important Questions 63

Question 84.
Evaluate \(\int \frac{\cos x+3 \sin x+7}{\cos x+\sin x+1} d x\)
Solution:
Let cos x + 3 sin x + 7 = A(cos x + sin x + 1)’ + B(cos x + sin x + 1) + C
Comparing the coefficients
A + B = 1, A – B = 3, B + C = 7
A = -1, B = 2, C = 5
Inter 2nd Year Maths 2B Integration Important Questions 64
Inter 2nd Year Maths 2B Integration Important Questions 65

Inter 2nd Year Maths 2B Integration Important Questions

Question 85.
Find \(\int \frac{x^{3}-2 x+3}{x^{2}+x-2}\) dx.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 66
Inter 2nd Year Maths 2B Integration Important Questions 67

Question 86.
Find \(\int \frac{d x}{x^{2}-81}\)
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 68

Inter 2nd Year Maths 2B Integration Important Questions

Question 87.
Find \(\int \frac{2 x^{2}-5 x+1}{x^{2}\left(x^{2}-1\right)}\) dx.
Solution:
Let \(\frac{2 x^{2}-5 x+1}{x^{2}\left(x^{2}-1\right)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-1}+\frac{D}{x+1}\)
2x2 – 5x + 1 = Ax(x2 – 1) + B(x2 – 1) + Cx2 (x + 1) + Dx2 (x – 1)
x = 1 ⇒ 2 – 5 + 1 = C (1 + 1) ⇒ 2C = -2
⇒ C = -1
x = -1 ⇒ 2 + 5 + 1 = D (-1 -1)
⇒ 8 = -2B ⇒ D = -4
x = 0 ⇒ 1 = B(-1) ⇒ B = -1
Equating the coefficients of x3
0 = A + C + D ⇒ A = -C – D = 1 + 4 = 5
Inter 2nd Year Maths 2B Integration Important Questions 69

Question 88.
Find \(\int \frac{3 x-5}{x\left(x^{2}+2 x+4\right)}\) dx.
Solution:
\(\frac{3 x-5}{x\left(x^{2}+2 x+4\right)}=\frac{A}{x}+\frac{B x+C}{x^{2}+2 x+4}\)
3x – 5 = A(x2 + 2x + 4) + (Bx + C) . x
x = 0 ⇒ -5 = 4 A ⇒ A = –\(\frac{5}{4}\)
Equating the coefficients of x2
A + B = 0 ⇒ B = -A = \(\frac{5}{4}\)
Equating the coefficient of x
3 = 2 A + C
C = 3 – 2 A = 3 + 2 . \(\frac{5}{4}\) = \(\frac{11}{2}\)
Inter 2nd Year Maths 2B Integration Important Questions 70
Inter 2nd Year Maths 2B Integration Important Questions 71

Inter 2nd Year Maths 2B Integration Important Questions

Question 89.
Find \(\int \frac{2 x+1}{x\left(x^{2}+4\right)^{2}} d x\)
Solution:
Let \(\frac{2 x+1}{x\left(x^{2}+4\right)^{2}}=\frac{A}{x}+\frac{B x+C}{x^{2}+4}+\frac{D x+E}{\left(x^{2}+4\right)^{2}}\)
2x + 1 = A (x2 + 4)2 + (Bx + C) + x (x2 + 4) + (Dx + E)x
Equating the coefficients of like power of x, we obtain
A + B = 0, C = 0, 8A + 4B + D = 0,
4C + E = 2, A = \(\frac{1}{16}\)
Solving these equation, we obtain
Inter 2nd Year Maths 2B Integration Important Questions 72
Inter 2nd Year Maths 2B Integration Important Questions 73

Question 90.
Evaluate \(\int\) x3 . e5x dx.
Solution:
We take a = 5, use the reduction formula
Inter 2nd Year Maths 2B Integration Important Questions 74
Inter 2nd Year Maths 2B Integration Important Questions 75

Question 91.
Evaluate \(\int\) sin4 x dx. [Mar. 14]
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 76
Inter 2nd Year Maths 2B Integration Important Questions 77

Inter 2nd Year Maths 2B Integration Important Questions

Question 92.
Evaluate \(\int\) tan6 x dx.
Solution:
Inter 2nd Year Maths 2B Integration Important Questions 78

Question 93.
\(\int\) sec5 x dx.
Solution:
Reduction formula is
Inter 2nd Year Maths 2B Integration Important Questions 79