CLASS 12 MATHS CHAPTER 7-INTEGRALS

Exercise 7.1

Solve The Following Questions.

Find an antiderivative (or integral) of the following functions by the method of inspection in Exercises 1 to 5.

Question 1. sin 2x
Solution :

The anti derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that,

herefore, the anti derivative of sin 2x is -1/2 cos 2x.

Question 2. cos 3x

Solution :

The anti derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that,

Therefore, the anti derivative of cos 3x is 1/3 sin 3x.

Question 3. e^2x

Solution :

The anti derivative of e^2x is the function of x whose derivative is e^2x.

It is known that,

Therefore, the anti derivative of  e^2x is 1/2 e^2x.

Question 4. (ax + b)²

Solution :

The anti derivative of (ax + b)² is the function of x whose derivative is (ax + b)².

It is known that,

Question 5. sin 2x – 4 e^3x

Solution :

The anti derivative of  sin 2x – 4 e^3x is the function of x whose derivative is sin 2x – 4 e^3x

It is known that,

Evaluate the following integrals in Exercises 6 to 11.
Question 6.  ∫(4e^3x+ 1) dx

Solution :
 
Question 7. 

Solution :

Question 8. 

Solution :


Question 9. 

Solution :

Question 10. 

Solution :

Question 11. 

Solution :

Evaluate the following integrals in Exercises 12 to 16.

Question 12. 

Solution :

Question 13. 

Solution :

Question 14. ∫(1 – x)√x dx

Solution : ∫(1 – x)√x dx

Question 15. 

Solution :

Question 16. 

Solution :

Evaluate the following integrals in Exercises 17 to 20.
Question 17. 

Solution :

 
Question 18.

Solution :

 
Question 19. 

Solution :

Question 20. 

Solution :

 

Question 21. Choose the correct answer:
The anti derivative of equals.


Solution :

Therefore, option (C) is correct.

Question 22. Choose the correct answer:

If   such that f(2) = 0 Then f is:


Solution :

It is given that,

Therefore, option (A) is correct.

Exercise 7.2

Solve The Following Questions.

Integrate the functions in Exercise 1 to 8.

Question 1.

Solution :

Let 1 + x² = t

∴2x dx = dt

Question 2. 

Solution :

Let log |x| = t

∴ 1/x dx = dt

Question 3. 

Solution :


Question 4. sin x ⋅ sin (cos x)

Solution :

sin x ⋅ sin (cos x)

Let cos x = t

∴ −sin x dx = dt

Question 5. sin(ax + b) cos(ax + b)

Solution :


Questionc 6. √ax + b

Solution :

Let ax + b = t

⇒ adx = dt

Question 7. x√x + 2

Solution :

Let  (x + 2) = t

∴ dx = dt

Question 8. x√1 + 2x²
Solution :

Let 1 + 2x² = t

∴ 4xdx = dt

Integrate the functions in Exercise 9 to 17.

Question 9.

Solution :
 

Question 10.

Solution :
 

Question 11.

Solution :


Question 12.

Solution :

Let x³ – 1 = t

∴ 3x² dx = dt

Question 13.

Solution :

Let 2 + 3x³ = t

∴ 9x² dx = dt

Question 14. 
Solution :

Let log x = t

∴ 1/x dx = dt

Question 15. x/9 – 4x²

Solution :

Let 9 – 4x² =  t

∴ −8x dx = dt

 

Question 16. 

Solution :

Let 2x + 3 = t

∴ 2dx = dt

Question 17. 
Solution :

Let x² = t

∴ 2xdx = dt

Integrate the functions in Exercise 18 to 26.

Question 18. 

Solution :

Question 19.

Solution :

Dividing numerator and denominator by e^x, we obtain

Question 20. 
Solution :

Question 21. tan² (2x – 3)

Solution :


Question 22. sec² (7 – 4x)

Solution :

Let 7 − 4x = t

∴ −4dx = dt

Question 23. 

Solution :

Question 24. 

Solution :

Question 25. 

Solution :


Question 26. 

Solution :

Let √x = t

Integrate the functions in Exercise 27 to 37.

Question 27.

Solution :

Let sin 2x = t

Question 28.

Solution :

Let 1 + sin x = t

∴ cos x dx = dt

Question 29. cot x log sin x

Solution :

Let log sin x = t

Question 30. sin x/1 + cos x

Solution :

Let 1 + cos x = t

∴ −sin x dx = dt

Question 31.  sin x/(1 + cos x)²

Solution :

Let 1 + cos x = t

∴ −sin x dx = dt

Question 32. 1/1 + cot x
Solution :
 

Question 33. 1/1 – tan x

Solution :

Question 34. 

Solution :
 
Question 35. 

Solution :

Let 1 + log x = t

∴ 1/x dx = dt

Question 36. 

Solution :

Question 37.

Solution :

Let x⁴ = t

∴ 4x³ dx = dt

Choose the correct answer in Exercise 38 and 39.

Question 38.  equals

(A) 10^x – x¹⁰ + C
(B) 10^x + x¹⁰ + C
(C) (10^x – x¹⁰)^-1 + C
(D) log(10^x + x¹⁰) + C

Solution :

Therefore, option (D) is correct.
Question 39.equals

(A) tan x + cot x + C
(B) tan x – cot x + C
(C) tan x cot x + C
(D) tan x – cot 2x + C

Solution :

Therefore, option (B) is correct.

Exercise 7.3

Solve The Following Questions.

Find the integrals of the following functions in Exercises 1 to 9.

Question 1. sin²(2x + 5)

Solution :


Question 2. sin 3x cos4x

Solution :


Question 3. cos 2x cos 4x cos 6x

Solution :


Question 4. sin³ (2x + 1)

Solution :


Question 5. sin³ x cos³ x

Solution :


Question 6. sin x sin 2x sin 3x

Solution :


Question 7. sin 4x sin 8x

Solution : It is known that,
sin A . sin B = 12cosA-B-cosA+B

∴∫sin4x sin8x dx=∫12cos4x-8x-cos4x+8xdx

=12∫cos-4x-cos12xdx

=12∫cos4x-cos12xdx

=12sin4x4-sin12x12+C

Question 8. 

Solution :


Question 9. 

Solution :


Find the integrals of the following functions in Exercises 10 to 18.

Question 10. sin⁴ x

Solution :

Question 11. cos⁴ 2x

Solution :


Question 12. 

Solution :


Question 13. 

Solution :


Question 14. 

Solution :


Question 15. tan³ 2x sec2x

Solution :

Question 16. tan⁴x

Solution :


Question 17. 

Solution :


Question 18. 

Solution :


Integrate the following functions in Exercises 19 to 22.
Question 19. 

Solution :

Question 20. 

Solution :
 
Question 21. sin⁻¹ (cos x)

Solution :

Question 22. 

Solution :


Choose the correct answer in Exercise 23 and 24.

Question 23.is equal to:

A. tan x + cot x + C

B. tan x + cosec x + C

C. − tan x + cot x + C

D. tan x + sec x + C

Solution :

Therefore, option (A) is correct.

Question 24. is equal to:

A. − cot (e^xx) + C

B. tan (xe^x) + C

C. tan (e^x) + C

D. cot (e^x) + C

Solution :
Let I = 

Let e^xx = t

Therefore, option (B) is correct.

Exercise 7.4

Solve The Following Questions.

Integrate the following functions in Exercises 1 to 9.

Question 1. 

Solution :

Let x³ = t

∴ 3x² dx = dt

Question 2. 

Solution :

Let 2x = t

∴ 2dx = dt

Question 3. 

Solution :

Let 2 − x = t

⇒ −dx = dt


Question 4. 

Solution :

Let 5x = t

∴ 5dx = dt

Question 5. 

Solution :
 

Question 6. 

Solution :

Let x³ = t

∴ 3x² dx = dt

Question 7. 

Solution :

Question 8. 

Solution :

Let x³ = t

∴ 3x² dx = dt

Question 9. 

Solution :

Let tan x = t

∴ sec²x dx = dt

Integrate the following functions in Exercises 10 to 18.

Question 10. 

Solution :


Question 11. 

Solution :


Question 12 .

Solution :


Question 13. 

Solution :


Question 14. 

Solution :


Question 15. 

Solution :


Question 16. 

Solution :

Question 17.

Solution :


Question 18. 

Solution :

Integrate the following functions in Exercises 19 to 23.
Question 19. 

Solution :

Question 20. 

Solution :

Question 21. 

Solution :

Using equations (2) and (3) in (1), we obtain

Question 22. 

Solution :

Question 23. 

Solution :

Choose the correct answer in Exercise 24 and 25.

Question 24.  equals

A. x tan⁻¹ (x + 1) + C

B. tan^{− 1} (x + 1) + C

C. (x + 1) tan⁻¹ x + C

D. tan⁻¹ x + C

Solution :


Therefore, option (B) is correct.

Question 25.  equals

Solution :


Therefore, option (B) is correct.

Exercise 7.5

Solve The Following Questions.

Integrate the (rational) function in Exercises 1 to 6.

Question 1. 

Solution :

Question 2. 

Solution :


Question 3. 

Solution :


Question 4. 

Solution :


Question 5.

Solution :


Question 6. 

Solution :

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (1 − x²) by x(1 − 2x), we obtain
 
Question7. 

Solution :


Question8. 

Solution :


Question9.  

Solution :


Question10. 

Solution :


Question11. 

Solution :


Question12. 

Solution :

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (x³ + x + 1) by x² − 1, we obtain

Integrate the following function in Exercises 13 to 17.

Question13. 

Solution :


Question 14. 

Solution :


Question15. 

Solution :


Question16.   [Hint: multiply numerator and denominator by x^{n − 1} and put xⁿ = t]

Solution :

Multiplying numerator and denominator by x^{n − 1}, we obtain



Question17.  [Hint: Put sin x = t]

Solution :


Integrate the following function in Exercises 18 to 21.

Question18. 

Solution :

Equating the coefficients of x³, x², x, and constant term, we obtain

A + C = 0

B + D = 4

4A + 3C = 0

4B + 3D = 10

On solving these equations, we obtain

A = 0, B = −2, C = 0, and D = 6

 

Question19. 

Solution :


Question20. 

Solution :

Multiplying numerator and denominator by x³, we obtain

Question21.  [Hint: Put e^x = t]

Solution :

 

Choose the correct answer in each of the Exercise 22 and 23.

Question22. equals:


Solution :

Therefore, option (B) is correct.

Question23. equals:

Solution :
 

Therefore, option (A) is correct.

Exercise 7.6

Solve The Following Questions.

Integrate the functions in Exercises 1 to 8.

Question 1. x sin x

Solution : Let I =  ∫ x sin x dx

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

Question 2. 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

Question 3. 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

Again integrating by parts, we obtain
 
Question 4. x logx

Solution : Let  I = ∫ x logx dx

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

Question 5. x log 2x

Solution : Let  I = ∫ x log 2x dx

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

Question 6. 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


Question 7. x sin^-1 x
Solution :
Let I = ∫ x sin^-1 x

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

Question 8. x tan^-1 x
Solution :
Let I = ∫ x tan^-1 x

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

Integrate the functions in Exercises 9 to 15.
Question 9. x cos^-1 x
Solution :
Let I = ∫ x cos^-1 x

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

Question 10. 
Solution :
Let I = ∫ .1 dx

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

Question 11. 
Solution :
Let  

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

Question 12. x sec² x
Solution : Let I = ∫ x sec² x dx

Taking x as first function and sec²x as second function and integrating by parts, we obtain

Question 13. tan^-1 x

Solution :
Let I = ∫ tan^-1 x dx

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

Question 14. x (log x)²
Solution :
Let I = ∫ x (log x)² dx

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

Question 15. (x² + 1) log x

Solution :

Integrate the functions in Exercises 16 to 22.
Question 16. 
Solution :

Question 17.
Solution :

Question 18. 
Solution :

Question 19. 
Solution :

Question 20. 
Solution :

Question 21. e^2x sin x
Solution :
Let I = ∫ e^2x sin x

Integrating by parts, we obtain

Question 22.
Solution :

Choose the correct answer in Exercise 23 and 24.
Question 23. equals to

Solution :
Let I = 

Therefore, option (A) is correct.
Question 24. ∫equals:
(A)  e^xcos x + C
(B) e^x sec x + C
(C) e^x sin x + C
(D) e^x tan x + C
Solution :

Therefore, option (B) is correct.

Exercise 7.7

Solve The Following Questions.

Integrate the functions in Exercises 1 to 9.

Question1.     

Solution :

Question2.    

Solution :

Question3. 

Solution :

Question4.    

Solution :

Question5.    

Solution :

Question6. 

Solution :

Question7.    

Solution :

Question8.    

Solution :

Question9. 

Solution :

Choose the correct answer in Exercise 10 to 11.

Question10.  is equal to:

Solution :

Therefore, option (A) is correct.

Question11. is equal to:

Therefore, option (D) is correct.

Exercise 7.8

Solve The Following Questions.

Evaluate the following definite integrals as limit of sums:

Question1.      

Solution :
It is know that 

Question2.    

Solution :
Let I = 

It is know that

Question3. 

Solution :
We know that

Question4.    

Solution :

We know that

From equations (2) and (3), we obtain

Question5.    

Solution : Let I = 
We know that

Question6. 

Solution :
We know that

Exercise 7.9

Solve The Following Questions.

Evaluate the definite integrals in Exercises 1 to 11.
Question1. 
Solution :

By second fundamental theorem of calculus, we obtain

Question2. 
Solution :

By second fundamental theorem of calculus, we obtain

I = F(3) – F(2)
= log|3| – log|2| = log 3/2

Question3. 
Solution :

By second fundamental theorem of calculus, we obtain

 
Question4. 
Solution :

By second fundamental theorem of calculus, we obtain

Question5. 
Solution :

By second fundamental theorem of calculus, we obtain

Question6. 
Solution :

By second fundamental theorem of calculus, we obtain

Question7. 
Solution :

By second fundamental theorem of calculus, we obtain

Question8. 
Solution :

By second fundamental theorem of calculus, we obtain

Question9. 
Solution :

By second fundamental theorem of calculus, we obtain

Question10. 
Solution :

By second fundamental theorem of calculus, we obtain

Question11. 
Solution :

By second fundamental theorem of calculus, we obtain

Evaluate the definite integrals in Exercises 12 to 20.
Question12. 
Solution :

By second fundamental theorem of calculus, we obtain

Question13. 
Solution :

By second fundamental theorem of calculus, we obtain

Question14. 
Solution :

Question15. 
Solution :

By second fundamental theorem of calculus, we obtain

Question16. 
Solution :

Equating the coefficients of x and constant term, we obtain

Question17. 
Solution :

By second fundamental theorem of calculus, we obtain

Question18. 
Solution :

By second fundamental theorem of calculus, we obtain

I = F(π) – F(0)

 = sin π – sin 0

 = 0
Question19. 
Solution :

By second fundamental theorem of calculus, we obtain

Question20. 
Solution :

By second fundamental theorem of calculus, we obtain

Choose the correct answer in Exercises 21 and 22.
Question21. equals:
(A) π/3
(B) 2π/3
(C) π/6
(D) π/12
Solution :

Therefore, option (D) is correct.
Question22. equals:
(A) π/6
(B) π/12
(C) π/24
(D) π/4
Solution :

By second fundamental theorem of calculus, we obtain

Therefore, option (C) is correct.

Miscellaneous Exercise

Solve The Following Questions.

Integrate the function in Exercises 1 to 11.
Question1.
Solution :

Question2. 
Solution :

Question3. 
Solution :

Takingθas first function and sec²θ as second function and integrating by parts, we obtain

Question4. 
Solution :

Question5. 
Solution :

Question6. 
Solution :

Question7. 
Solution :

Question8. 
Solution :

Question9. 

(A) 6

(B) 0

(C) 3

(D) 4
Solution :

Let cotθ = t ⇒ −cosec2θ dθ= dt

Question10. 

A. cos x + x sin x

B. x sin x

C. x cos x

D. sin x + x cos x

Solution :
Let I = 


Question11. 
Solution :

Integrate the function in Exercises 12 to 22.
Question12. 
Solution :


Question13. 
Solution :

Question14. 
Solution :

Question15. 
Solution :
Let I = 

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

Question16. 
Solution :
Let I = 

It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

Question17. 
Solution :

Question18. 
Solution :

Question19. 
Solution :

Question20. 
Solution :

Question21. 
Solution :
Let I = 
As sin² (−x) = (sin (−x))² = (−sin x)² = sin²x, therefore, sin²x is an even function.

Question22. 
Solution :


Evaluate the integrals in Exercises 23 and 24.
Question23. 
Solution :
Let I = 
As sin⁷ (−x) = (sin (−x))⁷ = (−sin x)⁷ = −sin⁷x, therefore, sin²x is an odd function.

Question24. 
Solution :

Evaluate the definite integrals in Exercise 25 to 33.
Question25. 
Solution :

Question26. 
Solution :

Adding (4) and (5), we obtain

Question27. 
Solution :

Question28. 
Solution :
Let I = 

It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

Question29.Show that  if f and g are defined as f (x) = f(a – x) and g(x) + g(a – x) = 4
Solution :

Question30. 

A. 0

B. 2

C. π

D. 1
Solution :

 = π
Question31. 

A. 2

B. 3/4

C. 0

D. -2

Solution :

Question32. 
Solution :

From equation (1), we obtain

Question33. 
Solution :


Prove the following (Exercise 34 to 40).
Question34.  [Hint: Put x = a/t]
Solution :

Question35. 
Solution :
Let I = 

Question36. 
Solution :

Question37. 
Solution :

Question38. 
Solution :

Question39. 
Solution :

Question 40. Evaluate as a limit of sum.
Solution :
Given: 

It is known that,

Question41. Choose the correct answer:is equal to:

Solution :

Therefore, option (A) is correct.

Question42. Choose the correct answer:is equal to:
(A) 
(B) log |sin x + cos x | + C
(C) log |sin x – cos x | + C
(D) 
Solution :

Therefore, option (B) is correct.

Question43. Choose the correct answers If f (a + b – x) = f (x), then 


Solution :

Therefore, option (D) is correct.
Question44. The value of is:
(A) 1
(B) 0
(C) -1
(D) π/4
Solution :

Therefore, option (B) is correct.