Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions Inter 2nd Year Maths 2B Integration Solutions Exercise 6(d) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2B Integration Solutions Exercise 6(d)

I. Evaluate the following integrals.

Question 1.
∫\(\frac{dx}{\sqrt{2x-3x^2+1}}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 1

Question 2.
∫\(\frac{\sin \theta}{\sqrt{2-\cos^2 \theta}}\)dθ
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 2

Question 3.
∫\(\frac{\cos x}{\sin^2 x+4sin x+5}\)dx
Solution:
t = sin x ⇒ dt = cos x dx
I = ∫\(\frac{dt}{t^2+4t+5}\) = ∫\(\frac{dt}{(t+2)^2+1}\)
= tan-1(t + 2) + C
= tan-1(sin x + 2) + C

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 4.
∫\(\frac{dx}{1+\cos^2 x}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 3

Question 5.
∫\(\frac{dx}{2\sin^2 x+3\cos^2 x}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 4

Question 6.
∫\(\frac{1}{1+\tan x}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 5

Question 7.
∫\(\frac{1}{1-\cot x}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 6

II. Evaluate the following integrals.

Question 1.
∫\(\sqrt{1+3x-x^2}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 7
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 8

Question 2.
∫\(\frac{9\cos x-\sin x}{4\sin x+5\cos x}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 9

Question 3.
∫\(\frac{2\cos x+3\sin x}{4\cos x+5\sin x}\)dx
Solution:
Let 2 cos c + 3 sin x = A(4 cos x + 5 sin x) + B(-4 sin x + 5 cos x)
Equating the co-efficient of sin x and cos x,
we get
4A + 5B = 2
5A – 4B = 3
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 10
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 11

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 4.
∫\(\frac{1}{1+\sin x+\cos x}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 12

Question 5.
∫\(\frac{1}{3x^2+x+1}\)dx
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 13

Question 6.
∫\(\frac{dx}{\sqrt{5-2x^2+4x}}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 14
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 15

III. Evaluate the following integrals.

Question 1.
∫\(\frac{x+1}{\sqrt{x^2-x+1}}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 16
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 17

Question 2.
∫(6x + 5)\(\sqrt{6-2x^2+x}\)dx
Solution:
Let 6x + 5 = A(1 – 4x) + B
Equating the constants
A + B = 5
B = 5 – A = 5 + \(\frac{3}{2}\) = \(\frac{13}{2}\)
∫(6x + 5)\(\sqrt{6-2x^2+x}\)dx
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 18
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 19

Question 3.
∫\(\frac{dx}{4+5\sin x}\)
Solution:
t = tan \(\frac{x}{2}\) ⇒ dt = sec² \(\frac{x}{2}\) . \(\frac{1}{2}\)dx
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 20

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 4.
∫\(\frac{1}{2-3\cos 2x}\)dx
Solution:
t = tan x ⇒ dt = sec² x dx
= (1 + tan² x)dx
= (1 +t²)dx
dx = \(\frac{dt}{1+t^2}\)
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 21

Question 5.
∫x\(\sqrt{1+x-x^2}\)dx
Solution:
Let x = A(1 – 2x) + B
Equating the coefficients of x
1 = -2 A ⇒ A = –\(\frac{1}{2}\)
Equating the constants
0 = A + B ⇒ B = -A = \(\frac{1}{2}\)
∫x\(\sqrt{1+x-x^2}\)dx
= –\(\frac{1}{2}\)∫(1 – 2x)\(\sqrt{1+x-x^2}\)dx + \(\frac{1}{2}\)∫\(\sqrt{1+x-x^2}\)dx
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 22

Question 6.
∫\(\frac{dx}{(1+x)\sqrt{3+2x-x^2}}\)
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 23
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 24

Question 7.
∫\(\frac{dx}{4\cos x+3\sin x}\)
Solution:
Let t = tan\(\frac{x}{2}\) so that dx = \(\frac{2dt}{1+t^2}\)
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 25

Question 8.
∫\(\frac{1}{\sin x+\sqrt{3} \cos x}\)dx
Solution:
Let t = tan \(\frac{x}{2}\) so that dx = \(\frac{2dt}{1+t^2}\)
sin x = \(\frac{2t}{1+t^2}\), cos x = \(\frac{1-t^2}{1+t^2}\)
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 26

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 9.
∫\(\frac{dx}{5+4\cos 2x}\).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 27

Question 10.
∫\(\frac{2\sin x+3\cos x+4}{3\sin x+4\cos x+5}\)dx.
Solution:
Let 2 sin x + 3 cos x + 4
= A(3 sin x + 4 cos x + 5) + 3(3 cos x – 4 sin x) + C
Equating the co-efficient of
sin x, we get 3A – 4B = 2
cos x, we get 4A + 3B = 3
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 28
Equating the constants
4 = 5A + C
C = 4 – 5A = 4 – 5.\(\frac{18}{25}\) = \(\frac{2}{5}\)
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 29
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 11.
∫\(\sqrt{\frac{5-x}{x-2}}\) dx on (2, 5).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 30
Let 5 – x = A. \(\frac{d}{dx}\)(7x – 10 – x²) + B
⇒ 5 – x = A(7 – 2x) + B
Equating coffs. of like terms
-2A = -1 ⇒ A = \(\frac{1}{2}\)
7A + B = 5
7(+\(\frac{1}{2}\)) + B = 5 ⇒ B = 5 – \(\frac{7}{2}\) = \(\frac{3}{2}\)
∴ 5 – x = \(\frac{1}{2}\)(7 – 2x) + \(\frac{3}{2}\)
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 31
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 32

Question 12.
∫\(\sqrt{\frac{1+x}{1-x}}\) dx on (-1, 1).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 33

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 13.
∫\(\frac{dx}{(1 – x)\sqrt{3-2x+x^2}}\) on (-1, 3).
Solution:
Put 1 – x = \(\frac{1}{1}\) ⇒ 1 – \(\frac{1}{1}\) = x\(\frac{1}{1-x}\) = t
dx = \(\frac{1}{t^2}\)dt
3 – 2x – x² = 3 – 2(\(\frac{1-1}{1}\)) – (\(\frac{1-1}{1}\))²
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 34
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 35

Question 14.
∫\(\frac{dx}{(x + 2)\sqrt{x+1}}\) on (-1, ∞).
Solution:
Put = x + 1 = t² ⇒ dx = 2t dt and
x + 2 = 1 + t²
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 36

Question 15.
∫\(\frac{dx}{(2x + 3)\sqrt{x+2}}\) on I ⊂ (-2, ∞)\{\(\frac{-3}{2}\)}.
Solution:
Put x + 2 = t² ⇒ dx = 2t dt and
2x + 3 = 2(t² – 2) + 3 = 2t² – 1
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 37

Question 16.
∫\(\frac{1}{(1+\sqrt{x}) \sqrt{x-x^2}}\)dx on (0, 1).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 38
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 39

Question 17.
∫\(\frac{dx}{(x + 1)\sqrt{2x^2+3x+1}}\) on
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 40
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 41

Question 18.
∫\(\sqrt{e^x-4}\) dx on [loge 4, ∞]
Solution:
Put ex – 4 = t² ⇒ ex dx = 2t dt
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 42

Question 19.
∫\(\sqrt{1+\sec x}\) dx on [(2n – \(\frac{1}{2}\))π – (2n + \(\frac{1}{2}\))π], (n ∈ Z).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 43
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 44

Inter 2nd Year Maths 2B Integration Solutions Ex 6(d)

Question 20.
∫\(\frac{dx}{1+x^4}\) on R.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 45
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 46
Inter 2nd Year Maths 2B Integration Solutions Ex 6(d) 47