NCERT Solutions for Chapter 7 Integrals Class 12 Maths Exercise 7.6

Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.6

Integrals Exercise 7.6 Solutions

1. Integrate the functions x sin x 

Solution

Let l = ∫x sin x dx 
Taking x as first function and sin x as second function and integrating by parts, we obtain 

= x(-cos x) - ∫1.(- cos x) dx
= -x cos x + sin x + C

2. Integrate the functions x sin 3x 

Solution

Let I = ∫x sin 3x dx 
Taking x as first function and sin 3x as second function and integrating by parts, we obtain 

3. Integrate the functions x² e^x.

Solution

Let I = ∫ x² e^x dx 
Taking x² as first function and e^x as second function and integrating by parts, we obtain 

= x² e^x - 2[xe^x - ∫e^x dx]
= x² e^x - 2[xe^x - e^x ]
= x² e^x - 2xe^x + 2e^x + C 
= e^x (x² - 2x + 2) + C

4. Integrate the functions x log x 

Solution

Let I = ∫ x log x dx 
Taking log x as first function and x as second function and integrating by parts, we obtain 

5. Integrate the functions x log 2x 

Solution

Taking log 2x as first function and x as second function and integrating by parts, we obtain 

6. Integrate the functions x² log x 

Solution

Let I = ∫x² log x dx 
Taking log x as first function and x² as second function and integrating by parts, we obtain 

7. Integrate the functions x sin^-1 x 

Solution

Taking sin^-1 x as first function and x as second function and integrating by parts, we obtain 

9. Integrate the functions x cos^-1 x 

Solution

Let I =  ∫x cos^-1 x dx 
Taking cos^-1 x as first function and x as second function and integrating by parts, we obtain 

10. Integrate the functions (sin^-1 x)² 

Solution

Let I = ∫(sin^-1 x)² .l dx 

Taking (sin^-1 x)² as first function and 1 as second function and integrating by parts, we obtain 

11. Integrate the functions (x cos^-1 x)/√(1 - x² ) . 

Solution

Taking cos^-1 x as first function and [-2x/√(1 - x²)] as second function and integrating by parts, we obtain 

12. Integrate the functions x sec² x.

Solution

Let I = ∫x sec² x dx

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

= x tan x -  ∫1 . tan x dx

= x tan x + log |cos x| + C 

13. Integrate the functions tan^-1 x.

Solution

Let I = ∫1. tan^-1 x dx 

Taking tan^-1 x as first function and 1 as second function and integrating by parts, we obtain 

14. Integrate the functions x (log x)² 

Solution

I = ∫ x (log x)² dx

Taking (log x)² as first function and x as second function and integrating by parts, we obtain 

15. Integrate the functions (x² + 1) log x

Solution

Let I = ∫(x² + 1) log x dx = ∫x² log x dx + ∫log x dx 

Let I = I₁ + I₂ …(1)

Where, I₁ = ∫x² log x dx and I₂ = ∫ log x dx 

I₁ = ∫x² log x dx

Taking log x as first function and x² as second function and integrating by parts, we obtain 

I₂ = ∫ log x dx 

Taking log x as first function and 1 as second function and integrating by parts, we obtain 

= x log x - ∫1 dx 
= x log x - x + C₂ ...(3) 
Using equation (2) and (3) in (1), we obtain 

16. Integrate the functions e^x (sinx + cosx)

Solution

Let I =  ∫e^x (sin x + cos x) dx 

Let f(x) = sin x

⇒ f'(x) = cos x

⇒ I = ∫ e^x {f(x) + f '(x)} dx 
It is known that,  ∫ e^x {f(x) + f '(x)} dx = e^x f(x) + C 
∴ I = e^x sin x + C 

17. Integrate the functions xe^x /(1 + x)² 

Solution

18. Integrate the function e^x [(1 + sinx)/(1 + cos x)] 

Solution

19. Integrate the functions e^x (1/x - 1/x² )

Solution 

Also, let 1/x = f(x) ⇒ f '(x) = -1/x² 

It is known that, ∫ e^x {f(x) + f '(x)}dx = e^x f(x) + C 

∴ I = e^x /x + C

20. Integrate the functions (x - 3)e^x /(x - 1)³ 

Solution

21. Integrate the functions e^2x sin x

Solution

Let I = ∫ e^2x sin x dx  ...(1) 
Integrating by parts, we obtain 

22. Integrate the functions sin^-1 [2x/(1 + x²)]

Solution

Let x = tan θ

⇒ dx = sec² θ dθ

23. Choose the correct answer ∫ x² e^x3 dx equals 

(A) (1/3) e^x3 + C 
(B) (1/3) e^x2 + C      

(C) (1/2) e^x3 + C 
(D) (1/3) e^x2 + C 

Solution

Let I = ∫ x² e^x3 dx 
Also, let x³ = t ⇒ 3x² dx = dt 

Hence, the correct answer is A. 

24. Choose the correct answer ∫ e^x sec x(1 + tan x) dx equals
(A) e^x cos x + C
(B) e^x sec x + C
(C) e^x sin x + C
(D) e^x tan x + C

Solution

∫ e^x sec x(1 + tan x) dx 
Let I = ∫ e^x sec x(1 + tan x) dx = ∫ e^x (sec x + sec x tan x) dx
Also, let sec x = f(x) ⇒ sec x tan x = f '(x) 

It is known that, ∫ e^x {f(x) + f '(x)} dx = e^x f(x) + C 
∴ I = e^x sec x + C 
Hence, the correct answer is B.