NCERT Solutions for Chapter 7 Integrals Class 12 Maths Exercise 7.11

Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.11

Integrals Exercise 7.11 Solutions

1. By using the properties of definite integrals, evaluate the integrals ∫₀^π/2 cos²x dx

Solution

2. By using the properties of definite integrals, evaluate the integrals  ∫₀^π/2 √(sin x)/(√sinx + √cos x)dx 

Solution

3. By using the properties of definite integrals, evaluate the integrals  ∫₀^π/2 (sin^3/2 x dx)/(sin^3/2 x + cos^3/2 x)

Solution

Adding (1) and (2), we obtain 

4. By using the properties of definite integrals, evaluate the integrals 

∫₀^π/2 (cos⁵ x dx)/(sin⁵x + cos⁵x)

Solution

5. By using the properties of definite integrals, evaluate the integrals 

∫_-5⁵ |x + 2| dx 

Solution

Let I =  ∫_-5⁵ |x + 2| dx 

It can be seen that (x + 2) ≤ 0 on [-5, -2] and (x + 2) ≥ 0 on [-2, 5]. 

6. By using the properties of definite integrals, evaluate the integrals 
∫₂⁸ |x - 5| dx 

Solution

Let I = ∫₂⁸ |x - 5| dx 
It can be seen that (x - 5) ≤ 0 on [2, 5] and (x - 5) ≥ 0 on [5, 8]. 

7. By using the properties of definite integrals, evaluate the integrals ∫₀⁴ x(1 - x)ⁿ  dx

Solution

8. By using the properties of definite integrals, evaluate the integrals ∫₀^π/4 log(1 + tan x) dx

Solution

9. By using the properties of definite integrals, evaluate the integrals ∫₀² x√(2 - x) dx 

Solution

10. By using the properties of definite integrals, evaluate the integrals ∫₀^π/2 (2 log sin x - log sin 2x) dx 

Solution

11. By using the properties of definite integrals, evaluate the integrals ∫_-π/2^π/2 sin² x dx 

Solution

As sin² (-x) = [sin (-x)]² = (-sin x)²  = sin² x , therefore, sin² x is an even function. 
It is known that if f(x) is an even function, then 

12. By using the properties of definite integrals, evaluate the integrals ∫₀^π xdx/(1 + sin x) 

Solution 

13. By using the properties of definite integrals, evaluate the integrals ∫_π/2^π/2 sin⁷x dx 

Solution

14. By using the properties of definite integrals, evaluate the integrals, ∫₀^2x cos⁵ x dx 

Solution

15. By using the properties of definite integrals, evaluate the integrals 

∫₀^π [(sin x - cos x)/(1 + sin x cos x)] dx

Solution 

Adding (1) and (2), we obtain 

⇒ l = 0 

16. By using the properties of definite integrals, evaluate the integrals ∫₀^π  log(1 + cos x) dx 

Solution

Let I = ∫₀^π  log(1 + cos x) dx ...(1) 

sin(π - x) = sin x

17. By using the properties of definite integrals, evaluate the integrals 

∫₀^a √x/[√x + (√a - x)] dx 

Solution

18. By using the properties of definite integrals, evaluate the integrals 
∫₀⁴ |x - 1| dx 

Solution 

I =  ∫₀⁴ |x - 1| dx 
It can be seen that, (x - 1) ≤ 0 when 0 ≤ x ≤ 1 and (x - 1) ≥ 0 when 1 ≤ x ≤ 4 

19. Show that ∫₀^a f(x) g(x) dx = 2 ∫₀^a f(x)  dx if f and g are defined as f(x) = f(a - x) and g(x) + g(a - x) = 4 

Solution

20. Choose the correct The value of  ∫_-π/2^π/2  (x³ + x cos x + tan⁵ x + 1) dx is 

(A) 0 
(B) 2 
(C) π
(D) 1 

Solution

21. Choose the correct The value of ∫₀^π/2 log  [(4 + 3 sinx)/(4 + 3 cos)] dx is 

(A) 2 
(B) 3/4
(C) 0 
(D) -2 

Solution

Hence, the correct answer is C.