Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions Inter 2nd Year Maths 2B Definite Integrals Solutions Exercise 7(b) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2B Definite Integrals Solutions Exercise 7(b)

I. Evaluate the following definite integrals.

Question 1.
\(\int_0^a(a^2x-x^3) d x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 1

Question 2.
\(\int_2^3 \frac{2 x}{1+x^2} d x\)
Solution:
I = [ln|1 + x²|]³2
= ln 10 – ln 5
= ln(10/5)
= ln 2

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 3.
\(\int_0^\pi \sqrt{2+2 \cos \theta} d \theta\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 2

Question 4.
\(\int_0^\pi\sin^3x\cos^3xd x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 3

Question 5.
\(\int_0^2|1-x|d x\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 4

Question 6.
\(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x\)
Solution:
\(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x dx}{1+e^x}\) ………….. (i)
cos x is even function
ex is neither even nor odd.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 5
Adding (i) and (ii) we get
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 6

Question 7.
\(\int_0^1\frac{dx}{\sqrt{3-2x}}\)
Solution:
3 – 2x = t²
-2dx = 2t dt
dx = -t dt
3 – (2.1) = t²
1 = t²
3 – 2.0 = t²

Question 8.
\(\int_0^a(\sqrt{a}-\sqrt{x})^2\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 7

Question 9.
\(\int_0^{\pi / 4} \sec^4\theta d\theta\)
Solution:
Let ∫sec4 θ dθ = \(\int_0^{\pi / 4} \sec^2\theta(1+\tan^2\theta)d\theta\)
Put tan θ = y
sec² θ dθ = dy
θ = \(\frac{\pi}{4}\) ⇒ y= 1
θ = 0 ⇒ y = 0
I = \(\int_0^a\)(1 + y²)dy = [y + \(\frac{y^3}{3}\)]¹0
= 1 + \(\frac{1}{3}=\frac{4}{3}\)

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 10.
\(\int_0^3\frac{x}{\sqrt{x^2+16}}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 8

Question 11.
I = \(\int_0^1\)x.e-x²dx
Solution:
I = \(\int_0^a\)x.e-x²dx
⇒ -x² = t
⇒ -2x dx = dt
2x dx = -dt
x = 1 ⇒ t = 0
x = 0 ⇒ t = 1
I = \(\frac{1}{2}\)\(\int_0^a\)-et dt
= \(\frac{1}{2}\)[-et]-10
= \(\frac{1}{2}\)[e0 – e-1]
= \(\frac{1}{2}\)(1 – \(\frac{1}{e}\))

Question 12.
I = \(\int_1^5\frac{dx}{\sqrt{2x-1}}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 9

II. Evaluate the following integrals:

Question 1.
I = \(\int_0^4\frac{x^2}{1+x}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 10

Question 2.
\(\int_{-1}^2 \frac{x^2}{x^2+2}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 11

Question 3.
I = \(\int_0^1 \frac{x^2}{x^2+2}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 12

Question 4.
\(\int_0^{\pi / 2}\)x² sin x dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 13

Question 5.
\(\int_0^4\)|2-x|dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 14

Question 6.
\(\int_0^{\pi / 2}\frac{\sin^5 x}{\sin^5 x+\cos^5 x}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 15
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 16

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 7.
\(\int_0^{\pi / 2}\frac{\sin^2 x-\cos^2 x}{\sin^3 x+\cos^3 x}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 17

Question 8.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 18
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 19

Question 9.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 20
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 21

Question 10.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 22
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 23
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 24

Question 11.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 25
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 26

Question 12.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 27
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 28

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 13.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 29
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 30

Question 14.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 31
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 32

Question 15.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 33
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 34

III. Evaluate the following integrals :

Question 1.
\(\int_0^{\pi / 2}\frac{dx}{4+5\cos x}\)
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 35

Question 2.
\(\int_a^b\sqrt{(x-a)(b-x)}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 36

Question 3.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 37
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 38

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 4.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 39
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 40

Question 5.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 41
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 42
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 43

Question 6.
\(\int_0^a\)x(a – x)ndx
Solution:
I = \(\int_0^a\)x(a – x)ndx ………… (i)
\(\int_0^a\)f(x) dx = \(\int_0^a\)f(a – x)dx
I = \(\int_0^a\)(a – x).(x)ndx ………… (ii)
I = \(\int_0^a\)axn dx – xn+1 dx
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 44

Question 7.
\(\int_0^2x\sqrt{2-x}\)dx
Solution:
I = \(\int_0^2x.\sqrt{2-x}\)dx
\(\int_0^2\)f(x)dx = \(\int_0^2\)f(a – x)dx
= \(\int_0^2\)(2 – x).√x dx
= \(\int_0^2\)((2√x – x√x)) dx
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 45

Question 8.
\(\int_0^{\pi}\)x sin³ x dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 46
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 47

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 9.
\(\int_0^{\pi}\frac{x}{1+\sin x}\)dx
Solution:
\(\int_0^{\pi}\frac{x}{1+\sin x}\)dx …………… (i)
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 48

Question 10.
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 49
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 50
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 51

Question 11.
\(\int_0^1\frac{log(1+x)}{1+x^2}\)dx
Solution:
Put x = tan θ
dx = sec² θ dθ
x = 0 ⇒ θ = 0
x = x ⇒ θ = \(\frac{\pi}{4}\)
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 52
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 53

Question 12.
\(\int_0^{\pi}\frac{x \sin x}{1+\cos^2 x}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 54

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 13.
\(\int_0^{\pi/2}\frac{\sin^2 x}{\cos x+\sin x}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 55

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 56

Question 14.
\(\int_0^{\pi}\frac{1}{3+2\cos x}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 57

Question 15.
\(\int_0^{\pi/4}\)log(1 + tan x)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 58

Question 16.
\(\int_{-1}^{3/2}\)|x sin πx| dx
Solution:
We know that |x. sin πx| = x . sin πx
where -1 ≤ x ≤ 1
and |x . sin πx| = – x.sin πx where 1 < x ≤ 3/2
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 59
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 60

Question 17.
\(\int_0^1sin^{-1}\frac{2x}{1+x^2}\)dx
Solution:
\(\int_0^1sin^{-1}\frac{2x}{1+x^2}\)dx
Put x = tan θ ⇒ dx = sec² θ dθ
x = 0 ⇒ θ = 0
x = 1 ⇒ θ π/4
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 61

Question 18.
\(\int_0^1\)x tan-1x dx
Solution:
\(\int_0^1\)x. tan-1x dx
Put x = tan θ ⇒ dx = sec² θ dθ
x = 0 ⇒ θ = 0;
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 62

Question 19.
\(\int_0^{\pi}\frac{x\sin x}{1+\cos^2 x}\)dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 63

Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b)

Question 20.
Suppose that f : R → R is continuous periodic function and T is the period of it. Let a ∈ R. Then prove that for any positive integer n,
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 64
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(b) 65