Integrals Class 12th Mathematics Part Ii CBSE Solution

Class 12^th 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

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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 x² is 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.

e^2x

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 e^3x

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 – 4e^3x

Therefore, the anti-derivative of sin 2x – 4 e^3x 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, 

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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.

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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: 

Question 10.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: