NCERT Solutions for Chapter 7 Integrals Class 12 Maths Exercise 7.10

Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.10

Integrals Exercise 7.10 Solutions

1. Evaluate the integrals in using substitution
∫₀¹ x/(x² + 1) dx 

Solution

Let x² + 1 = t ⇒ 2x dx = dt 
when x = 0, t = 1 and when x = 1, t = 2 

2. Evaluate the integrals in using substitution 

∫₀^π/2 √(sin Ø) cos⁵ф dф 

Solution

= 64/231

3. ∫₀¹ sin^-1 [2x/(1 + x²)] dx 

Solution

Consider, I = ∫₀¹ sin^-1 [2x/(1 + x² )] dx 

Let x = tan θ ⇒ dx = sec² θ dθ
When x = 0, θ = 0 and when x = 1, θ = π/4 

Taking θ as first function and sec² θ as second function and integrating by parts, we obtain 

4. Evaluate the integrals in using substitution ∫₀² [x√(x + 2)] (Put x + 2 = t²) 

Solution

∫₀² [x√(x + 2)]
Let x + 2 = t² ⇒ dx = 2tdt 
When x = 0, t = √2 and when x = 2, t = 2 

5. Evaluate the integrals in using substitution ∫₀^π/2 sin x/(1 + cos² x) dx 

Solution

∫₀^π/2 sin x/(1 + cos² x) dx 
Let cos x = t ⇒ -sin x dx = dt 
when x = 0, t = 1 and when x = π/2, t = 0 

6. Evaluate the integrals in using substitution  ∫₀² dx/(x + 4 - x²)

Solution

Let x - 1/2 = t ⇒ dx = dt 
When x = 0, t = -1/2 and when x = 2, t = 3/2 

7. Evaluate the integrals in using substitution ∫_-1¹ dx/(x² + 2x + 5)

Solution

Let x + 1 = t ⇒ dx = dt 
When x = -1, t = 0 and when x = 1, t = 2 

8. Evaluate the integrals in using substitution ∫₁² (1/x - 1/2x²) e^2x dx 

Solution

Let 2x = t ⇒ 2dx = dt 
When x = 1, t = 2 and when x = 2, t = 4 

9. The value of the integral  ∫_1/3⁴ [(x - x³ )^1/3]/x⁴ dx is 

(A) 6 
(B) 0
(C) 3 
(D) 4 

Solution

Let cot θ = t ⇒ - cosec² θ dθ = dt 
When θ = sin^-1 (1/3) . t = 2√2 and when θ = π/2 . t = 0 

Hence, the correct answer is A. 

10. If f(x) = ∫₀^x t sin t dt, then f'(x) is
(A) cos x + x sin x

(B) x sin x

(C) x cos x

(D) sin x + x cos x

Solution

f(x) = ∫₀^x t sin t dt

Integrating by parts, we obtain 

= - x cos x + sin x

⇒ f'(x) = -[{x(-sin x)} + cos x] + cos x 

= x sin x - cos x + cos x 

= x sin x

Hence, the correct answer is B.