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

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

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

I. Evaluate the following definite integrals.

Question 1.
\(\int_{\pi/2}^{\pi/2}\)sin10 x dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 1

Question 2.
\(\int_0^{\pi/2}\)cos11 x dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 2

Question 3.
\(\int_0^{\pi/2}\)cos7 x . sin²x dx.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 3

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

Question 4.
\(\int_0^{\pi/2}\)sin4 x . cos4 x dx.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 4

Question 5.
\(\int_0^{\pi/2}\)sin³ x cos6 x dx.
Solution:
\(\int_0^{\pi/2}\)sin³ x cos6 x dx.
\(\int_0^{\pi/2}\)(1 – cos² x) cos6 x.sin x dx
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 5

Question 6.
\(\int_0^{2\pi}\)sin² x cos4 x dx.
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 6

Question 7.
\(\int_{-\pi/2}^{\pi/2}\)sin² θ cos7 θ dθ.
Solution:
sin² θ cos7 θ is even function
f(θ) = sin² θ . cos7 θ dθ
f(-θ) = sin² (-θ) . cos7 (-θ)
= f(θ)
= 2\(\int_{-\pi/2}^{\pi/2}\)sin² θ cos7 θ dθ

\(\int_0^{\pi/2}\)sinm x cosnx dx
n is odd n = 7
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 7

Question 8.
\(\int_{-\pi/2}^{\pi/2}\)sin³ θ .cos³ θ dθ.
Solution:
f(θ) = sin³ θ . cos³ θ dθ
f(-θ) = sin³ (-θ) . cos³ (-θ)
= -sin³ θ cos³ θ = -f(θ)
f(θ) is odd
∴ \(\int_{-\pi/2}^{\pi/2}\)sin³ θ.cos³ θ dθ = 0

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

Question 9.
\(\int_0^a\)x(a² – x²)7/2 dx
Solution:
x = a sin θ, a = a sin θ
dx = a cos θ dθ, θ = π/2
= \(\int_0^{\pi/2}\)a sin θ(a² – a²sin²θ)7/2 a cos θ dθ
= \(\int_0^{\pi/2}\)a9 cos8 θ sin θ dθ
= a9\(\int_0^{\pi/2}\)cos8 . θ sin θ dθ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 8

Question 10.
\(\int_0^2\)x3/2.\(\sqrt{2-x}\)dx
Solution:
x = 2 cos² θ
dx = 4 cos θ sin θ dθ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 9

II. Evaluate the following integrals.

Question 1.
\(\int_0^1\)x5(1 – x)3/2 dx
Solution:
x = sin² θ
dx = 2 sin θ . cos θ . dθ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 10

Question 2.
\(\int_0^4\)(16 – x²)5/2 dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 11

Question 3.
\(\int_{-3}^3\)(9 – x²)3/2 x dx
Solution:
Let f(x) = (9 – x²)3/2x
f(x) = (9 – (-x²))3/2(-x)
= (9 – x²)3/2 . x
= -f(x)
∴ f is odd function
∴ \(\int_{-3}^3\)(9 – x²)3/2 x dx = 0

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

Question 4.
\(\int_0^5\)x³(25 + x²)7/2 dx
Solution:
Let I = \(\int_0^5\)x³(25 + x²)7/2 dx
Put x = 5 sin θ
dx = 5 cosθ dθ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 12
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 13

Question 5.
\(\int_{-\pi}^{\pi}\)sin8 x cos7 x dx
Solution:
Let f(x) = sin8 x. cos7 x
f(-x) = sin8 (-x) . cos7 (-x)
= sin8 x. cos7 x
∴ f is even function.
∴ \(\int_{-\pi}^{\pi}\)sin8 x cos7 x dx = 2\(\int_0^{\pi}\)sin8 x cos7 x = 0

Question 6.
\(\int_3^7 \sqrt{\frac{7-x}{x-3}}\)dx
Solution:
Put x = 3 cos²θ + 7 sin²θ
dx = (7 – 3)sin2θ dθ
dx = 4 sin 2θ dθ
U.L.
x = 3 cos²θ + 7 sin²θ
7 = 3 cos²θ + 7 sin²θ
4 cos²θ = 0
θ = \(\frac{\pi}{2}\)
L.L
x = 3 cos²θ + 7 sin²θ
3 = 3 sin²θ + 7 sin²θ
4 sin²θ = 0
sinθ = 0
θ = 0
7 – x = 7 – (3 cos²θ + 7 sin²θ)
= (7 – 3)cos²θ
= 4 cos²θ
x – 3 = 3 cos²θ + 7 sin²θ – 3
= (7 – 3)sin²θ
= 4 sin²θ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 14

Question 7.
\(\int_2^6\sqrt{(6-x)(x-2)}\)dx
Solution:
Put x = 2 cos²θ + 6 sin²θ
dx = (6 – 2) sin2θ dθ
dx = 4 sin2θ dθ
U.L
x = 2 cos²θ + 6 sin²θ
6 = 2 cos²θ + 6 sin²θ
4 cos²θ = 0
cos θ = 0
θ = \(\frac{\pi}{2}\)

L.L
x = 2 cos²θ + 6 sin²θ
2 = 2 cos²θ + 6 sin²θ
4 sin²θ = 0
θ = 0
6 – x = 6 – (2 cos²θ + 6 sin²θ)
= (6 – 2) cos²θ
= 4 cos²θ
x – 2 = 2 cos²θ + 6 sin²θ – 2
= (6 – 2)sin²θ
= 4 sin²θ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 15

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

Question 8.
\(\int_0^{\pi}\)tan5x cos8x dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 16

III. Evaluate the following integrals.

Question 1.
\(\int_0^1\)x7/2 (1 – x)5/2 dx
Solution:
Put x = sin²θ
dx = 2 sin θ cos θ dθ
U.L
x = sin²θ
1 = sin²θ
θ = \(\frac{\pi}{2}\)

L.L
x = sin²θ
0 = sin²θ
θ = 0
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 17

Question 2.
\(\int_0^{\pi}\)(1 + cos x)³ dx
Solution:
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 18
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 19

Question 3.
\(\int_4^9\frac{dx}{\sqrt{(9 – x)(x – 4)}}\)
Solution:
Put x = 4 cos²θ + 9 sin²θ
dx = (9 – 4)sin2θ dθ
dx = 5 sin2θ dθ

U.L
x = 4 cos²θ + 9 sin²θ
9 = 4 cos²θ + 9 sin²θ
5 cos²θ = 0
θ = \(\frac{\pi}{2}\)

L.L
x = 4 cos²θ + 9 sin²θ
4 = 4 cos²θ + 9 sin²θ
5 sin²θ = 0
θ = 0

9 – x = 9 – (4 cos²θ + 9 sin²θ)
= (9 – 4) cos²θ
= 5 cos²θ

x – 4 = 4 cos²θ + 9 sin²θ – 4
= (9 – 4) sin²θ
= 5 sin²θ
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 20
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 21

Question 4.
\(\int_0^5\)x²(\(\sqrt{5-x}^7\) dx
Solution:
Put x = 5 sin²θ
dx = 10 sinθ cosθ dθ

U.L
x = 5 sin²θ
5 = 5 sin²θ
sin²θ = 1
θ = \(\frac{\pi}{2}\)

L.L
x = 5 sin²θ
0 = 5sin²θ
sin²θ = 0
θ = 0
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 22

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

Question 5.
\(\int_0^{2\pi}\)(1 + cos x)5(1 – cos x)³ dx.
Solution:
\(\int_0^{2\pi}\)(1 + cos x)5(1 – cos x)³ dx . (1 + cos x)³(1 + cos x)²(1 – cos x)³
Inter 2nd Year Maths 2B Definite Integrals Solutions Ex 7(c) 23