Integrals Class 12th Mathematics Part Ii CBSE Solution

Class 12th Mathematics Part Ii CBSE Solution
Exercise 7.1
  1. sin 2x Find an anti-derivative (or integral) of the following functions by the…
  2. cos 3x Find an anti-derivative (or integral) of the following functions by the…
  3. e2x Find an anti-derivative (or integral) of the following functions by the…
  4. (ax + b)^2 Find an anti-derivative (or integral) of the following functions by…
  5. sin 2x - 4 e3x Find an anti-derivative (or integral) of the following functions…
  6. Find the following integrals. integrate (4e^3x + 1) dx
  7. integrate x^2 (1- 1/x^2) dx Find the following integrals.
  8. integrate (ax^2 + bx+c) dx Find the following integrals.
  9. integrate 2x^2 + e^x dx Find the following integrals.
  10. integrate (root x - 1/root x)^2 dx Find the following integrals.
  11. integrate x^3 + 5x^2 - 4/x^2 dx Find the following integrals.
  12. integrate x^3 + 3x+4/root x dx Find the following integrals.
  13. integrate x^3 - x^2 + x-1/x-1dx Find the following integrals.
  14. integrate (1-x) root xdx Find the following integrals.
  15. integrate root x (3x^2 + 2x+3) dx Find the following integrals.
  16. integrate (2x-3cosx+e^x) dx Find the following integrals.
  17. integrate (2x^2 - 3sinx+5 root x) dx Find the following integrals.…
  18. integrate secx (secx+tanx) dx Find the following integrals.
  19. integrate sec^2x/cosec^2xdx Find the following integrals.
  20. integrate 2-3sinx/cos^2xdx Find the following integrals.
  21. The anti-derivative of (root x + 1/root x) equalsA. 1/3 x^1/3 + 2x^1/2 + c B.…
  22. If d/dx f (x) = 4x^3 - 3/x^4 such that f(2) = 0. Then f(x) isA. x^4 + 1/x^3 -…
Exercise 7.10
  1. Evaluate the integrals using substitution. integrate _0^1 x/x^2 + 1 dx…
  2. integrate _0^pi /2 root sinphi cos^5phi d phi Evaluate the integrals using…
  3. integrate _0^1sin^-1 (2x/1+x^2) dx Evaluate the integrals using substitution.…
  4. integrate _0^2root x+2 (putx+2 = t^2) Evaluate the integrals using substitution.…
  5. integrate _0^pi /2 sinx/1+cos^2xdx Evaluate the integrals using substitution.…
  6. integrate _0^2 dx/x+4-x^2 Evaluate the integrals using substitution.…
  7. integrate _-1^1 dx/x^2 + 2x+5 Evaluate the integrals using substitution.…
  8. integrate _1^2 (1/x - 1/2x^2) e^2x dx Evaluate the integrals using substitution.…
  9. The value of the integral integrate _ 1/3^1 (x-x^3)^1/3/x^4 dx isA. 6 B. 0 C. 3…
  10. If f (x) = integrate _0^xtsintegrate dt then f'(x) isA. B. xsinx C. xcosx D.…
Exercise 7.11
  1. integrate _0^pi /2 cos^2xdx By using the properties of definite integrals,…
  2. integrate _0^pi /2 root sinx/root sinx + root cosx dx By using the properties of…
  3. integrate _0^pi /2 sin^3/2 xdx/sin^3/2 x+cos^3/2 x dx By using the properties of…
  4. integrate _0^pi /2 cos^5xdx/sin^5x+cos^5x By using the properties of definite…
  5. integrate _-5^5 |x+2|dx By using the properties of definite integrals, evaluate…
  6. integrate _2^8 |x-5|dx By using the properties of definite integrals, evaluate…
  7. integrate _0^1z (1-x)^n dx By using the properties of definite integrals,…
  8. integrate _0^pi /4 log (1+tanx) dx By using the properties of definite…
  9. integrate _0^2x root 2-xdx By using the properties of definite integrals,…
  10. integrate _0^pi /2 (2logsinx-logsin2x) dx By using the properties of definite…
  11. integrate _ - pi /2^pi /2 sin^2xdx By using the properties of definite…
  12. integrate _0^pi xdx/1+sinx By using the properties of definite integrals,…
  13. integrate _ - pi /2^pi /2 sin^7xdx By using the properties of definite…
  14. By using the properties of definite integrals, evaluate the integrals integrate…
  15. integrate _0^pi /2 sinx-cosx/1+sinxcosxdx By using the properties of definite…
  16. integrate _0^pi log (1+cosx) dx By using the properties of definite integrals,…
  17. integrate _0^a root x/root x + root a-x dx By using the properties of definite…
  18. integrate _0^4 |x-1|dx By using the properties of definite integrals, evaluate…
  19. Show that integrate _0^af (x) g (x) dx = 2 integrate _0^af (x) dx_s if f and g…
  20. The value of integrate _ - pi /2^pi /2 (x^3 + xcosx+tan^3x+1) dx A. 0 B. 2 C.…
  21. The value of integrate _0^pi /2 log (4+3sinx/4+3cosx) dxjs A. 2 B. 3/4 C. 0 D.…
Miscellaneous Exercise
  1. Integrate the function: 1/x-x^3
  2. Integrate the function: 1/root x+a + root x+b
  3. Integrate the function: 1/x root ax-x^2 [: putx = a/t]
  4. Integrate the function: 1/x^2 (x^4 + 1)^3/4
  5. Integrate the function: 1/x^1/2 + x^1/3 [hintegrate 1/x^1/2 + x^1/3 = 1/x^1/3…
  6. Integrate the function: 5x/(x+1) (x^2 + 9)
  7. Integrate the function: sinx/sin (x-a)
  8. Integrate the function: e^5logx-e^4logx/e^3logx-e^2logx
  9. Integrate the function: cosx/root 4-sin^2x
  10. Integrate the function:
  11. Integrate the function: 1/cos (x+a) cos (x+b)
  12. Integrate the function: x^3/root 1-x^8
  13. Integrate the function: e^x/(1+e^x) (2+e^x)
  14. Integrate the function: 1/(x^2 + 1) (x^2 + 4)
  15. Integrate the function: cos^3xe^logsinx
  16. Integrate the function: e^3logx (x^4 + 1)^-1
  17. Integrate the function: f^there there eξ sts (ax+b) [f (ax+b)]^n
  18. Integrate the function: 1/root sin^3xsin (x + alpha)
  19. Integrate the function: sin^-1root x-cos^-1root x/sin^-1root x+cos^-1root x , x…
  20. Integrate the function: root 1 - root x/1 + root x
  21. Integrate the function: 2+sin2x/1+cos2xe^x
  22. Integrate the function: x^2 + x+1/(x+1)^2 (x+2)
  23. Integrate the function: tan^-1root 1-x/1+x
  24. Integrate:
  25. Evaluate the definite integral: integrate _ pi /2^pi e^x (1-sinx/1-cosx) dx…
  26. Evaluate the definite integral: integrate _0^pi /4 sinxcosx/cos^4x+sin^4xdx…
  27. Evaluate the definite integral: integrate _0^pi /2 cos^2xdx/cos^2x+4sin^2x…
  28. Evaluate the definite integral: integrate _ pi /6^pi /3 sinx+cosx/root sin2x dx…
  29. Evaluate the definite integral: integrate _0^1 dx/root 1+x - root x…
  30. Evaluate the definite integral: integrate _0^pi /4 sinx+cosx/9+16sin2xdx…
  31. Evaluate the definite integral: integrate _0^pi /2 sin2xtan^-1 (sinx) dx…
  32. Evaluate the definite integral: integrate _0^pi xtanx/secx+tanxdx…
  33. Evaluate the definite integral: integrate _1^4 [|x-1|+|x-2|+|x-3|]dx…
  34. Prove: integrate _1^3 dx/x^2 (x+1) = 2/3 + log 2/3
  35. Prove: integrate _0^1xe^x dx = 1
  36. Prove: integrate _-1^1x^17cos^4xdx = 0
  37. Prove: integrate _0^pi /2 sin^3xdx = 2/3
  38. Prove: integrate _0^pi /4 2tan^3xdx = 1-log2
  39. Prove: integrate _0^1sin^-1xdx = pi /2 - 1
  40. Evaluate integrate _0^1e^2-3x dx as a limit of a sum.
  41. integrate dx/e^x + e^-x is equal to Choose the correct answersA. log (e^x -…
  42. integrate cos2x/(sinx+cosx)^2 dx is equal to Choose the correct answersA.…
  43. If f (a + b - x) = f (x), then integrate _a^bxf (x) is equal to Choose the…
  44. The value of integrate _0^1tan^-1 (2x-1/1+x-x^2) dx Choose the correct…
Exercise 7.2
  1. 2x/1+x^2 Integrate the functions.
  2. (logx)^2/x Integrate the functions.
  3. 1/x+xlogx Integrate the functions.
  4. sin x sin (cos x) Integrate the functions.
  5. Integrate the functions. sin (ax + b) cos (ax + b)
  6. root ax+b Integrate the functions.
  7. x root x+2 Integrate the functions.
  8. x root 1+2x^2 Integrate the functions.
  9. (4x+2) root x^2 +x+1 Integrate the functions.
  10. 1/x - root x Integrate the functions.
  11. x/root x+4 , x0 Integrate the functions.
  12. (x^3 - 1)^1/x x^5 Integrate the functions.
  13. x^2/(2+3x^3)^3 Integrate the functions.
  14. 1/x (logx)^m , x0 , m not equal 1 Integrate the functions.
  15. x/9-4x^2 Integrate the functions.
  16. e^2x+3 Integrate the functions.
  17. x/e^x^2 Integrate the functions.
  18. e^tan^-1x/1+x^2 Integrate the functions.
  19. e^2x - 1/e^2x + 1 Integrate the functions.
  20. e^2x - e^-2x/e^2x + e^-2x Integrate the functions.
  21. tan^2 (2x - 3) Integrate the functions.
  22. sec^2 (7 - 4x) Integrate the functions.
  23. sin^-1x/root 1-x^2 Integrate the functions.
  24. 2cosx-3sinx/6cosx+4sinx Integrate the functions.
  25. 1/cos^2x (1-tanx)^2 Integrate the functions.
  26. cosroot x/root x Integrate the functions.
  27. root sin2x cos2x Integrate the functions.
  28. cosx/root 1+sinx Integrate the functions.
  29. cot x log sin x Integrate the functions.
  30. sinx/1+cosx Integrate the functions.
  31. sinx/(1+cosx)^2 Integrate the functions.
  32. 1/1+cotx Integrate the functions.
  33. 1/1-tanx Integrate the functions.
  34. root tanx/sinxcosx Integrate the functions.
  35. (1+logx)^2/x Integrate the functions.
  36. (x+1) (x+logx)^2/x Integrate the functions.
  37. x^3sin (tan^-1x^4)/1+x^8 Integrate the functions.
  38. Choose the correct answer: A. 10x x^10 + C B. 10x + x^10 + C C. (10x x^10)1 + C…
  39. integrate dx/sin^2xcos^2x equalsA. tan x + cot x + C B. tan x - cot x + C C.…
Exercise 7.3
  1. sin^2 (2x+5) Find the integrals of the functions.
  2. cos2xcos4xcos6x Find the integrals of the functions.
  3. sin3xcos4x Find the integrals of the functions.
  4. sin^3 (2x+1) Find the integrals of the functions.
  5. sin^3xcos^3x Find the integrals of the functions.
  6. sinxsin2xsin3x Find the integrals of the functions.
  7. sin4xsin8x Find the integrals of the functions.
  8. 1-cos/1+cos Find the integrals of the functions.
  9. cosx/1+cosx Find the integrals of the functions.
  10. sin^4x Find the integrals of the functions.
  11. cos^42x Find the integrals of the functions.
  12. sin^2x/1+cosx Find the integrals of the functions.
  13. cos2x-cos2alpha /cosx-cosalpha Find the integrals of the functions.…
  14. cosx-sinx/1+sin2x Find the integrals of the functions.
  15. tan^32xsec2x Find the integrals of the functions.
  16. tan^4x Find the integrals of the functions.
  17. sin^3x+cos^3x/sin^2xcos^2x Find the integrals of the functions.
  18. cos2x+2sin^2x/cos^2x Find the integrals of the functions.
  19. 1/sinxcos^3x Find the integrals of the functions.
  20. cos2x/(cosx+sinx)^2 Find the integrals of the functions.
  21. sin^-1 (cosx) Find the integrals of the functions.
  22. 1/cos (x-a) cos (x-b) Find the integrals of the functions.
  23. integrate sin^2x-cos^2x/sin^2xcos^2xdx A. tanx+cotx+c B. tanx+cosecx+c C.…
  24. integrate e^x (1+x)/cos^2 (e^xx) A. -cot (ex^4) + c B. tan (e^x) + c C. tan…
Exercise 7.4
  1. 3x^2/x^6 + 1 Integrate the functions.
  2. 1/root 1+4x^2 Integrate the functions.
  3. 1/root (2-x)^2 + 1 Integrate the functions.
  4. 1/root 9-25x^2 Integrate the functions.
  5. 3x/1+2x^4 Integrate the functions.
  6. x^2/1-x^6 Integrate the functions.
  7. x-1/root x^2 - 1 Integrate the functions.
  8. x^2/root x^6 + a^6 Integrate the functions.
  9. sec^2x/root tan^2x+4 Integrate the functions.
  10. 1/root x^2 + 2x+2 Integrate the functions.
  11. 1/9x^2 + 6x+5 Integrate the functions.
  12. 1/root 7-6x-x^2 Integrate the functions.
  13. 1/root (x-1) (x-2) Integrate the functions.
  14. 1/root 8+3x-x^2 Integrate the functions.
  15. 1/root (x-a) (x-b) Integrate the functions.
  16. 4x+1/root 2x^2 + x-3 Integrate the functions.
  17. x+2/root x^2 - 1 Integrate the functions.
  18. 5x-2/1+2x+3x^2 Integrate the functions.
  19. 6x+7/root (x-5) (x-4) Integrate the functions.
  20. x+2/root 4x-x^2 Integrate the functions.
  21. x+2/root x^2 + 2x+3 Integrate the functions.
  22. x+3/x^2 - 2x-5 Integrate the functions.
  23. 5x+3/root x^2 + 4x+10 Integrate the functions.
  24. Choose the correct answer: integrate dx/x^2 + 2x+2 A. xtan^-1 (x+1) + c B.…
  25. Choose the correct answer: integrate dx/root 9x-4x^2 A. 1/9 sin^-1 (9x-8/8) + c…
Exercise 7.5
  1. x/(x+1) (x+2) Integrate the rational functions.
  2. 1/x^2 - 9 Integrate the rational functions.
  3. 3x-1/(x-1) (x-2) (x-3) Integrate the rational functions.
  4. x/(x-1) (x-2) (x-3) Integrate the rational functions.
  5. 2x/x^2 + 3x+2 Integrate the rational functions.
  6. 1-x^2/x (1-2x) Integrate the rational functions.
  7. x/(x^2 + 1) (x-1) Integrate the rational functions.
  8. x/(x-1)^2 (x+2) Integrate the rational functions.
  9. 3x+5/x^3 - x^2 - x+1 Integrate the rational functions.
  10. 2x-3/(x^2 - 1) (2x+3) Integrate the rational functions.
  11. 5x/(x+1) (x^2 - 4) Integrate the rational functions.
  12. x^3 + x+1/x^2 - 1 Integrate the rational functions.
  13. 2/(1-x) (1+x^2) Integrate the rational functions.
  14. 3x-1/(x+2)^2 Integrate the rational functions.
  15. 1/x^4 - 1 Integrate the rational functions.
  16. 1/x (x^n + 1) [Hint: multiply numerator and denominator by xn-1 and put xn=t]…
  17. cosx/(1-sinx) (2-sinx) [: x = t] Integrate the rational functions.…
  18. (x^2 + 1) (x^2 + 2)/(x^2 + 3) (x^2 + 4) Integrate the rational functions.…
  19. 2x/(x^2 + 1) (x^2 + 3) Integrate the rational functions.
  20. 1/x (x^4 - 1) Integrate the rational functions.
  21. 1/(6^x - 1) [Hint: Put ex = t] Integrate the rational functions.
  22. Choose the correct answer integrate xdx/(x-1) (x-2) A. log| (x-1)^2/x-2|+c B.…
  23. Integrate the rational functions.A. log|x| - 1/2 log (x^2 + 1) + c B. log|x| +…
Exercise 7.6
  1. xsinx Integrate the functions.
  2. xsin3x Integrate the functions.
  3. x^2e^x Integrate the functions.
  4. xlogx Integrate the functions.
  5. xlog2x Integrate the functions.
  6. x^2logx Integrate the functions.
  7. xsin^-1x Integrate the functions.
  8. xtan^-1x Integrate the functions.
  9. xcos^-1x Integrate the functions.
  10. (sin^-1x)^2 Integrate the functions.
  11. xcos^-1x/root 1-x^2 Integrate the functions.
  12. xsec^2x Integrate the functions.
  13. tan^-1x Integrate the functions.
  14. x (logx)^2 Integrate the functions.
  15. (x^2 + 1) logx Integrate the functions.
  16. e^x (sinx+cosx) Integrate the functions.
  17. xe^x/(1+x)^2 Integrate the functions.
  18. e^x (1+sinx/1+cosx) Integrate the functions.
  19. e^x (1/x - 1/x^2) Integrate the functions.
  20. (x-3) e^x/(x-1)^3 Integrate the functions.
  21. e^2xsinx Integrate the functions.
  22. sin^-1 (2x/1+x^2) Integrate the functions.
  23. integrate x^2e^x^3 dxequdls Choose the correct answer:A. 1/3 e^x^3 + c B. 1/3…
  24. integrate e^xsecx (1+tanx) dx Choose the correct answer:A. e^xcosx+c B.…
Exercise 7.7
  1. Integrate : root 4-x^2
  2. Integrate : root 1-4x^2
  3. Integrate : root x^2 + 4x+6
  4. Integrate : root x^2 + 4x+1
  5. Integrate : root 1-4x-x^2
  6. Integrate : root x^2 + 4x-5
  7. Integrate : root 1+3x-x^2
  8. Integrate : root x^2 + 3x
  9. Integrate : root 1 + x^2/9
  10. integrate root 1+x^2 dx is equal to Choose the correct answerA. x/2 root 1+x^2…
  11. integrate root x^2 - 8x+7 dx is equal to A. 1/2 (x-4) root x^2 - 2x+7 +…
Exercise 7.8
  1. integrate _a^bxdx Evaluate the following definite integrals as limit of sums.…
  2. integrate _0^5 (x+1) dx Evaluate the following definite integrals as limit of…
  3. integrate _2^3x^2 dx Evaluate the following definite integrals as limit of sums.…
  4. integrate _1^4 (x^2 - x) dx Evaluate the following definite integrals as limit…
  5. integrate _-1^1e^x dx Evaluate the following definite integrals as limit of…
  6. integrate _0^4 (x+e^2x) dx Evaluate the following definite integrals as limit of…
Exercise 7.9
  1. Evaluate integrate _-1^1 (x+1) dx
  2. Evaluate integrate _2^3 1/x dx
  3. Evaluate integrate _1^2 (4x^3 -5x^2 +6x+9) dx
  4. Evaluate integrate _0^pi /4 sin2xdx
  5. Evaluate integrate _0^pi /2 cos2xdx
  6. Evaluate integrate _4^5e^x dx
  7. Evaluate integrate _0^pi /4 tanxdx
  8. Evaluate integrate _ pi /6^pi /4 cosecxdx
  9. Evaluate integrate _0^1 dx/root 1-x^2
  10. Evaluate integrate _0^1 dx/1+x^2
  11. Evaluate integrate _2^3 dx/x^2 - 1
  12. Evaluate integrate _0^pi /2 cos^2xdx
  13. Evaluate integrate _2^3 xdx/x^2 + 1
  14. Evaluate integrate _0^1 2x+3/5x^2 + 1 dx
  15. Evaluate integrate _0^1xe^x^2 dx
  16. Evaluate integrate _1^2 5x^2/x^2 + 4x+3
  17. Evaluate integrate _0^pi /1 (2sec^2x+x^3 + 2) dx
  18. Evaluate integrate _0^pi (sin^2 x/2 - cos^2 x/2) dx
  19. Evaluate integrate _0^2 6x+3/x^2 + 4 dx
  20. Evaluate integrate _0^1 (xe^x + sin pi x/4) dx
  21. integrate _1^root 3 dx/x^2 + 1 equalsA. π/3 B. 2π/3 C. π/6 D. π/12…
  22. integrate dx/4+9x^2 equalsA. π/6 B. π/12 C. π/24 D. π/4

Exercise 7.1
Question 1.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x


Answer:

Method: To find the anti derivative of a function by inspection 
Steps: 1. In this method we look for a function whose derivative is the given function. For Example: if we need to find anti derivative of x2, we know that derivative of x3 is 3x2. Therefore, the variable terms comes out to be the same.
2. After that balance out the coefficients of variables by dividing and multiplying suitable terms. From above example if [we divide x3 by 3 we will get the answer as x2. Hence, we can say that anti derivative of xis x3/3.

Now, similarly,

We know that 


Therefore, the anti-derivative of sin2x is 


Question 2.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

cos 3x


Answer:

We know that 


Therefore, the anti-derivative of cos3x is .



Question 3.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

e2x


Answer:

We know that 


Therefore, the anti-derivative of  is .



Question 4.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

(ax + b)2


Answer:

We know that 


Therefore, the anti-derivative of  is .



Question 5.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x – 4 e3x


Answer:

We know that



Therefore, the anti-derivative of sin2x is  …(1)


Also,


Therefore, the anti-derivative of  is  …(2)


From (1) and (2), we get,



= sin 2x – 4e3x


Therefore, the anti-derivative of sin 2x – 4 e3x is .


Question 6.

Find the following integrals.



Answer:




Question 7.

Find the following integrals.



Answer:






Question 8.

Find the following integrals.



Answer:






Question 9.

Find the following integrals.



Answer:






Question 10.

Find the following integrals.



Answer:


Now we know that,

∫xn dx 

Therefore,


Question 11.

Find the following integrals.



Answer:



Separating the terms we get,

Applying the formula,
∫xn dx = 




Answer.
 
Question 12.

Find the following integrals.



Answer:


Separating the terms we get,

Applying the formula,

∫ xn dx = 




Question 13.

Find the following integrals.



Answer:



Now the numerator can be factorized as,

x3 - x2 + x - 1 = x2(x - 1) + 1(x - 1)

x3 - x2 + x - 1 = (x2 + 1)(x - 1)

Now putting this in given integral we get,




Question 14.

Find the following integrals.



Answer:





Question 15.

Find the following integrals.



Answer:





Question 16.

Find the following integrals.



Answer:




Question 17.

Find the following integrals.



Answer:





Question 18.

Find the following integrals.



Answer:


Formulas Used: ∫ sec2x dx = tanx + c and ∫ secx tanx dx = sec x + c

Opening the brackets we get,


Answer.
 
Question 19.

Find the following integrals.



Answer:







= tanx –x +C



Question 20.

Find the following integrals.



Answer:




= 2tanx – 3secx + C



Question 21.

The anti-derivative of  equals
A. 

B. 

C. 

D. 


Answer:



Question 22.

If  such that f(2) = 0. Then f(x) is
A.  B. 

C.  D. 

Solution ||| The correct option is (A).


Answer:

It is given that 



Also, It is given that f(2) = 0




Therefore, 



Exercise 7.10
Question 1.

Evaluate the integrals using substitution.



Answer:

Given: 

Let x2 + 1 = t


⇒ 2xdx = dt


⇒ xdx = � dt


When x = 0, t = 1 and when x = 1, t = 2








Question 2.

Evaluate the integrals using substitution.



Answer:

Given: 

Let 




Also, let 


when,


so,








Question 3.

Evaluate the integrals using substitution.



Answer:

Given: 

Let x = tan θ ⇒ dx = sec2 θ d θ


When, x = 0, θ = 0 and when x = 1, θ = π /4


Let






By applying product rule as,











Question 4.

Evaluate the integrals using substitution.



Answer:

Given: 

Let x + 2 = t2 ⇒ dx = 2t dt


And x = t2 -2


when, x = 0, t = √2 and when x = 2, t = 2


so,













Question 5.

Evaluate the integrals using substitution.



Answer:

Given: 

Let cos x = t


⇒ -sin xdx = dt


⇒ sin xdx = -dt


When x = 0, t = 1 and when x = π /2, t = 0



because, 








Question 6.

Evaluate the integrals using substitution.



Answer:

Given: 


we can write it as,




let


when



because,















Question 7.

Evaluate the integrals using substitution.



Answer:

Given: 



Let x + 1 = t


⇒ dx = dt


When x = -1, t = 0 and when x = 1, t = 2



because,






Question 8.

Evaluate the integrals using substitution.



Answer:

Given: 

Let 2x = t ⇒ 2 dx = dt


When x = 1, t = 2 and when x = 2, t = 4






now, let 1/t = f(t)


then, f' (t) = -1/t2



because,








Question 9.

The value of the integral  is
A. 6

B. 0

C. 3

D. 4


Answer:

Given: 

let 


Now, let x = sin θ ⇒ dx = cos θ d θ


when, and when x = 1, θ = π/2









Now, let cot θ = t ⇒ - cosec2 θ d θ


when,









= 6


Correct option is: (A)


Question 10.

If  then f'(x) is
A. 

B. 

C. 

D. 


Answer:

Given: 

Applying product rule,


⇒ 





= -x cos x + sin x – 0


⇒ f(x) = -x cos x + sinx



⇒ f'(x) = -[{x(-sinx )} + cosx ] + cosx


= x sin x – cos x + cos x


= x sin x


Correct answer is B.



Exercise 7.11
Question 1.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 

let,


as,




Adding (1) and (2), we get









Question 2.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 

let,


as,




Adding (1) and (2), we get






Question 3.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 





Adding (1) and (2), we get









Question 4.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 





Adding (1) and (2), we get









Question 5.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 

As we can see that (x+2) ≤ 0 on [-5, -2] and (x+2) ≥ 0 on [-2,5]










Question 6.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 

As we can see that (x-5)≤0 on [2,5] and (x+2)≥0 on [5,8]









⇒ I = 9



Question 7.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 











Question 8.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: 














Question 9.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given: