423 + Interview Questions in INTEGRATION IN GENERAL AWARENESS Page 1 InterviewSolution

1.	Find the value of the integrates given below :∫(sec2x + cosec2x)dx
Answer» ∫(sec²x + cosec²x)dx = ∫sec²xdx + ∫cosec²xdx = tanx - cotx + c

1.

Find the value of the integrates given below :∫(sec2x + cosec2x)dx

Answer»

∫(sec²x + cosec²x)dx

= ∫sec²xdx + ∫cosec²xdx

= tanx - cotx + c

Discussion

2.	Find the value of the integrates given below :∫sec x (sec x + tan x) dx
Answer» ∫sec x (sec x + tan x) dx = ∫sec² x dx + ∫sec x tan x dx = tan x + sec x + C

2.

Find the value of the integrates given below :∫sec x (sec x + tan x) dx

Answer»

∫sec x (sec x + tan x) dx

= ∫sec² x dx + ∫sec x tan x dx

= tan x + sec x + C

Discussion

3.	Find the value of the integrates given below :∫ logx x dx
Answer» ∫ log_x x dx = \(\int\frac{logx}{log_ex}dx\) = ∫1dx = x + c

3.

Find the value of the integrates given below :∫ logx x dx

Answer»

∫ log_x x dx

= \(\int\frac{logx}{log_ex}dx\)

= ∫1dx = x + c

Discussion

4.	Find the value of the integrates given below :∫ cot x (tan x – cosec x) dx
Answer» ∫ cot x (tan x – cosec x) dx = ∫ cot x tan x dx – ∫ cot x cosec x dx = ∫ 1 dx – ∫ cosec x cot x dx = x + cosec x + C

4.

Find the value of the integrates given below :∫ cot x (tan x – cosec x) dx

Answer»

∫ cot x (tan x – cosec x) dx

= ∫ cot x tan x dx – ∫ cot x cosec x dx

= ∫ 1 dx – ∫ cosec x cot x dx

= x + cosec x + C

Discussion

5.	Find the value of the integrates given below :∫ (tan2 x – cot2 x) dx
Answer» ∫ (tan² x – cot² x) dx = ∫ (sec² x -1 – cosec² x +1) dx = ∫ sec² x dx – ∫ cosec² x dx = tan x + cot x + C

5.

Find the value of the integrates given below :∫ (tan2 x – cot2 x) dx

Answer»

∫ (tan² x – cot² x) dx

= ∫ (sec² x -1 – cosec² x +1) dx

= ∫ sec² x dx – ∫ cosec² x dx

= tan x + cot x + C

Discussion

6.	Find the value of the integrates given below :∫ cot2 x dx
Answer» ∫ (cosec² x – 1) dx = ∫ cosec² x dx – ∫dx = – cot x – x + C

6.

Find the value of the integrates given below :∫ cot2 x dx

Answer»

∫ (cosec² x – 1) dx

= ∫ cosec² x dx – ∫dx

= – cot x – x + C

Discussion

7.	Find the value of the integrates given below :∫(5cosx - 3sinx + 2/cos2x)dx
Answer» ∫(5cosx - 3sinx + 2/cos²x)dx = 5∫cosxdx - 3∫sinxdx + 2∫sec²xdx = 5sinx - 3(-cosx) + 2tanx + c = 5sinx + 3cosx + 2tanx + c

7.

Find the value of the integrates given below :∫(5cosx - 3sinx + 2/cos2x)dx

Answer»

∫(5cosx - 3sinx + 2/cos²x)dx

= 5∫cosxdx - 3∫sinxdx + 2∫sec²xdx

= 5sinx - 3(-cosx) + 2tanx + c

= 5sinx + 3cosx + 2tanx + c

Discussion

8.	`int_(0)^(pi//4) cos 0. " cosec"^(2) 0 do`
Answer» Correct Answer - `(2-sqrt(2))`

Discussion

9.	`int_(0)^(pi//2) (dx)/(1+2 cos x)`
Answer» Correct Answer - `(1)/(sqrt(3)) log (2+sqrt(3))`

Discussion

10.	`int_(0)^(pi//2) x sin cos x dx=?`A. `(pi)/(4)`B. `(pi)/(8)`C. `(pi)/(12)`D.
Answer» Correct Answer - C

Discussion

11.	`int_(0)^(pi) x sin x. cos^(2) x dx`
Answer» Correct Answer - `(pi)/(3)`

Discussion

12.	`int_(0)^(pi//2) x sin x cos x dx`
Answer» Correct Answer - `(pi)/(8)`

Discussion

13.	Evaluate the following integral: `int_1^3(cos(logx))/x dx`
Answer» Correct Answer - ` sin (log 3)`

Discussion

14.	`int_(0)^(pi//4) sqrt(cot x dx) =?`A. `(Pi sqrt(2))/(4)+(1)/(sqrt(2)) log (sqrt(2)-1)`B. `(-pisqrt(2))/(4)-(1)/(sqrt(2)) log (sqrt(2)-1)`C. `(pisqrt(2))/(4) -(1)/(sqrt(2))log (sqrt(2)-1)`D.
Answer» Correct Answer - D

Discussion

15.	`int_(0)^(1//2) (x sin^(-1)x)/(sqrt(1-x^(2)))dx=?`A. `(pi sqrt(3))/(12)+(1)/(2)`B. `(pisqrt(3))/(12)-(1)/(2)`C. `(pisqrt(3))/(12) -(1)/(2)`D.
Answer» Correct Answer - A

Discussion

16.	`int_(0)^(1) sqrt((1-x)/(1+x)) dx=?`A. `(pi)/(2)+1`B. `(pi)/(2) -1`C. None of theseD.
Answer» Correct Answer - C

Discussion

17.	`int_(0)^(1) (x)/(sqrt(1+x^(2)))dx=?`A. `sqrt(2)-1`B. `sqrt(2)`C. `-sqrt(2)`D.
Answer» Correct Answer - B

Discussion

18.	`int_(0)^(1) x sqrt((1-x^(2))/(1+x^(2)))dx`
Answer» Correct Answer - `((pi)/(4)-(1)/(2))`

Discussion

19.	`intsqrt((x+1)/(x-1))dx`
Answer» Correct Answer - `sin^(-1) x-sqrt(1-x^(2)) +c`

Discussion

20.	Evaluate `int(1)/(1+sinx)dx`.
Answer» Correct Answer - `sqrt(2) log \|tan ((pi)/(8)+(x)/(4))\|+c`

Discussion

21.	`(i) (log x. sin[1+(log x)^(2)])/(x) dx` `(ii) int(dx)/(x(1+log)^(n))`
Answer» Correct Answer - `(i) -(1)/(2) cos [1+(log x)^(2)]+c " "(ii) (1)/(-n+1) (1+log x)^(-n+1) +c`

Discussion

22.	`int1/(x^2(x^4+1)^(3/4))dx`
Answer» Correct Answer - `-(1)/(x) (x^(4) +1)^(1//4) +c`

Discussion

23.	`intsqrt(1+sin 2x) dx`
Answer» Correct Answer - `sin x-cos x+c`

Discussion

24.	`int(x^(2) +3)/(x^(2)+1)dx`
Answer» Correct Answer - `x+2 tan^(-1) x+c`

Discussion

25.	`int" cos"^(3)(3x +5) dx`
Answer» Correct Answer - `(1)/(3) sin (3x +5) -(1)/(9) sin^(3) (3x+5) +c`

Discussion

26.	`int(2x^(3))/((x^(2)+1)^(2))dx`
Answer» Correct Answer - `log (x^(2)+1) +(1)/(x^(2) +1)+c` first write 2x^3 as 2x.x^2 then derivation of x^2 is 2x which is in numerator now put x^2+1 that is denominator eaualt to t then differentiate w.r.t t then solve ques

Discussion

27.	Evaluate: `int1/(sin(x-a)sin(x-b)) dx`
Answer» Correct Answer - `(1)/(sin (a-b)) log \|(sin (x-a))/(sin(x-b))\|+c`

Discussion

28.	Let `f(x)=(sin^(2)pix)/(1+pi^(x)).` Then, `int[f(x)+f(-x)]dx` is equal to
Answer» Correct Answer - B (b) `f(x)+f(-x)=(sin^(2)pix)/(1+pi^(x))+(sin^(2)(-pix))/(1+pi^(-x))` `=(sin^(2)pix)/(1+pi^(x))+((sin^(2)pix)pi^(x))/(pi^(x)+1)` `=sin^(2)pix=(1-cos 2pix)/(2)` `therefore int[f(x)+f(-x)]dx=int[(1-cos 2pix)/(2)]dx` `=(1)/(2)xx-(sin 2pix)/(4pi)+C`

Discussion

29.	`int(x+1)(x+2)^7(x+3)dx` is equal toA. `((x+2)^(10))/(10)-((x+2)^(8))+C`B. `((x+1)^(2))/(2)-((x+2)^(8))/(8)-((x+3)^(2))/(2)+C`C. `((x+2)^(10))/(10)+C`D. `((x+1)^(2))/(2)+((x+2)^(8))/(8)+((x+3)^(2))/(2)+C`
Answer» Correct Answer - A (a) `(x+1)(x+3)=(x+2-1)(x+2-1)` `=(x+1)^(2)-1` `therefore int(x+1)(x+2)^(7)(x+3)dx=int{(x+2)^(9)-(x+2)^(7)}dx=((x+2)^(10))/(10)-((x+2)^(8))/(8)+C`

Discussion

30.	` int(1)/(5+4 cos x)dx`
Answer» Correct Answer - `(2)/(3) tan^(-1) .((1)/(3) tan. (x)/(2))+c`

Discussion

31.	`(i)intsec^(7) x. sin x dx" "(ii) int(1)/(sinx. cos^(2) x)dx`
Answer» Correct Answer - `(i) (1)/(6) sec^(6) x+c " "(ii) sec x+ log \| " cosec " x- cot x \|+c`

Discussion

32.	`int(1)/((asin x+b cos x)^(2))dx`
Answer» Correct Answer - `(-1)/(a(a tan x+b))+c`

Discussion

33.	The value of `int(x^(2)+1)/(x^(4)-x^(2)+1)dx` isA. `tan^(-1)(2x^(2)-1)+C`B. `tan^(-1)(x^(2)+1)/(x)+C`C. `sin^(-1)(x-(1)/(x))+C`D. `tan^(-1)((x^(2)-1)/(x))+C`
Answer» Correct Answer - D Let `l=int(x^(2)+1)/(x^(4)-x^(2)+1)dx=int(1+(1)/(x^(2)))/(x^(2)+(1)/(x^(2))-1)dx` `l=((1+(1)/(x^(2))))/((x-(1)/(x))^(2)+1)dx` `"Put "x-(1)/(x)=t rArr (1+(1)/(x^(2)))dx=dt` `therefore" "l=int(dt)/(t^(2)+1)tan^(-1)t+C = tan^(-1)(x-(1)/(x))+C` `=tan^(-1)((x^(2)-1)/(x))+C`

Discussion

34.	Evaluate: (i) `int1/(x^2)cos^2(1/x) dx`(ii) `intsec^4xtanx dx`
Answer» Correct Answer - `(1)/(4) sec^(4) x+c`

Discussion

35.	`int" cos"^(4) " 2x dx "`
Answer» Correct Answer - `(3x)/(8) +(1)/(8) sin 4x+ +(1)/(64) sin 8x +c`

Discussion

36.	`int(1)/(sqrt(x^(2) -9))dx`
Answer» Correct Answer - `log \| x+ sqrt(x^(2) -9) "\|"+c`

Discussion

37.	If `(d(f(x))/(dx)=(1)/(1+x^(2)) `then `(d)/(dx){f(x^(3))} `isA. `(3x)/(1+x^(3))`B. `(3x^(2))/(1+x^(6))`C. `(-6x^(5))/((1+x^(6))^(2))`D. `(-6x^(5))/(1+x^(6))`
Answer» Correct Answer - B Given, `(d)/(dx){f(x)}=(1)/(1+x^(2))` On integrating both sides, we get `f(x)=tan^(-1)x` `therefore" "(d)/(dx)f(x^(3))=(d)/(dx)(tan^(-1)x^(3))` `=(1)/(1+(x^(3))^(2)).3x^(2)=(3x^(2))/(1+x^(6))`

Discussion

38.	`int ((x+2)/(x+4))^2 e^x dx` is equal toA. `e^(x)((x)/(x+4))+C`B. `e^(x)((x+2)/(x+4))+C`C. `e^(x)((x-2)/(x+4))+C`D. `((2xe^(x))/(x+4))+C`
Answer» Correct Answer - A Let `l=int((x+2)/(x+4))^(2)e^(x)dx=inte^(x)[(x^(2)+4+4x)/((x+4)^(2))]dx` `rArr" "l=int e^(x)[(x(x+4))/((x+4)^(2))+(4)/((x+4)^(2))]dx` `rArr" "l=int (e^(x)x)/(x+4)dx+int(4e^(x))/((x+4)^(2))dx` On using integration by parts, we get `rArr" "l=e^(x)((x)/(x+4))-int(4e^(x))/((x+4)^(2))dx+int(4e^(x))/((x+4)^(2))dx` `rArr" "l=(xe^(x))/((x+4))+C`

Discussion

39.	`int(1)/(2+sin^(2) x)dx`
Answer» Correct Answer - `(1)/(sqrt(6))tan^(-1) ((sqrt(3) tanx)/(sqrt(2)))+c`

Discussion

40.	`int(1)/(4x^(2) +9) dx`
Answer» Correct Answer - `(1)/(6) tan^(-1) ((2x)/(3))+c`

Discussion

41.	`int(2x-1)/(sqrt(x^(2)-x-1))dx`
Answer» Correct Answer - `2sqrt(x^(2) -x-1)+c`

Discussion

42.	`int(dx)/(cos x-sinx)` is equal toA. `(1)/(sqrt2)log\|tan((x)/(2)-(pi)/(8))\|+C`B. `(1)/(sqrt2)log\|cot((x)/(2))\|+C`C. `(1)/(sqrt2)log\|tan((x)/(2)-(3pi)/(8))\|+C`D. `(1)/(sqrt2)log\|tan((x)/(2)+(3pi)/(8))\|+C`
Answer» Correct Answer - D `l=int(dx)/(cosx-sinx)` `=(1)/(sqrt2)int(dx)/(((1)/(sqrt2)cosx-(1)/(sqrt2)sinx))` `=(1)/(sqrt2)int(dx)/(cos(x+(pi)/(4)))` `=(1)/(sqrt2)intsec(x+(pi)/(4))dx` `=(1)/(sqrt2)log\|tan((pi)/(4)+(x)/(2)+(pi)/(8))\|+C` `=(1)/(sqrt2)log\|tan((x)/(2)+(3pi)/(8))\|+C`

Discussion

43.	`int(x^2dx)/(4+x^2)`
Answer» Correct Answer - `x-2 tan^(-1) .(x)/(2)+c`

Discussion

44.	`(i) int sin 2x. cos5x dx " "(ii) int(sin 4x)/(sin x) dx`
Answer» Correct Answer - `(i) -(1)/(14) cos 7x +(1)/(7) cos7x +(1)/(6) cos 3x +c " "(ii) (2)/(3) sin 3x +2sin x+c`

Discussion

45.	`int(x^(2) +1)/((x+1))dx`
Answer» Correct Answer - `x-2 log 9(x+1) -(2)/(x+1)+c`

Discussion

46.	`int cos 2x . cos 4x . cos 6x dx`
Answer» Correct Answer - `(x)/(4)+(1)/(16) sin 4 x+(1)/(32) sin 8x +(1)/(48) sin12x +c`

Discussion

47.	`int(sin^(6)x+cos^(6)x+3sin^(2)x cos^(2)x)dx` is equal toA. `x+C`B. `(3)/(2)sin2x+C`C. `-(3)/(2)cos 2x+C`D. `(1)/(3)sin 3x -cos 3x+C`
Answer» Correct Answer - A Let `l=int(sin^(6)x+cos^(6)x+3sin^(2)x cos^(2)x)dx` `=int{(sin^(2)x)^(3)+(cos^(2)x)^(3)+3sin^(2)x cos^(2)x}dx` `=int[{:((sin^(2)x+cos^(2)x)"("sin^(4)x+cos^(4)x),(-sin^(2)xcos^(2)x")"+3sin^(2)x cos^(2)x):}]` `=int[{:((sin^(2)x+cos^(2)x)^(2)-3sin^(2)x cos^(2)x),(" "+3sin^(2)xcos^(2)x):}]` `=int1dx=x+C`

Discussion

48.	`int(1)/(sqrt(5-.(x^(2))/(4)))dx`
Answer» Correct Answer - `2sin^(-1) ((x)/(2sqrt(5)))+c`

Discussion

49.	`int sin^(2//3) x cos^(3) x dx`
Answer» Correct Answer - `(3)/(5)(sinx)^(5//3)-(3)/(11)(sinx)^(11//3) +c`

Discussion

50.	`intsec^(4) x dx`
Answer» Correct Answer - `tan x+(1)/(3) tan^(3) x+c`

Discussion

Explore topic-wise InterviewSolutions in .

Find the value of the integrates given below :∫(sec2x + cosec2x)dx

Find the value of the integrates given below :∫sec x (sec x + tan x) dx

Find the value of the integrates given below :∫ logx x dx

Find the value of the integrates given below :∫ cot x (tan x – cosec x) dx

Find the value of the integrates given below :∫ (tan2 x – cot2 x) dx

Find the value of the integrates given below :∫ cot2 x dx

Find the value of the integrates given below :∫(5cosx - 3sinx + 2/cos2x)dx

`int_(0)^(pi//4) cos 0. " cosec"^(2) 0 do`

`int_(0)^(pi//2) (dx)/(1+2 cos x)`

`int_(0)^(pi//2) x sin cos x dx=?`A. `(pi)/(4)`B. `(pi)/(8)`C. `(pi)/(12)`D.

`int_(0)^(pi) x sin x. cos^(2) x dx`

`int_(0)^(pi//2) x sin x cos x dx`

Evaluate the following integral: `int_1^3(cos(logx))/x dx`

`int_(0)^(pi//4) sqrt(cot x dx) =?`A. `(Pi sqrt(2))/(4)+(1)/(sqrt(2)) log (sqrt(2)-1)`B. `(-pisqrt(2))/(4)-(1)/(sqrt(2)) log (sqrt(2)-1)`C. `(pisqrt(2))/(4) -(1)/(sqrt(2))log (sqrt(2)-1)`D.

`int_(0)^(1//2) (x sin^(-1)x)/(sqrt(1-x^(2)))dx=?`A. `(pi sqrt(3))/(12)+(1)/(2)`B. `(pisqrt(3))/(12)-(1)/(2)`C. `(pisqrt(3))/(12) -(1)/(2)`D.

`int_(0)^(1) sqrt((1-x)/(1+x)) dx=?`A. `(pi)/(2)+1`B. `(pi)/(2) -1`C. None of theseD.

`int_(0)^(1) (x)/(sqrt(1+x^(2)))dx=?`A. `sqrt(2)-1`B. `sqrt(2)`C. `-sqrt(2)`D.

`int_(0)^(1) x sqrt((1-x^(2))/(1+x^(2)))dx`

`intsqrt((x+1)/(x-1))dx`

Evaluate `int(1)/(1+sinx)dx`.

`(i) (log x. sin[1+(log x)^(2)])/(x) dx` `(ii) int(dx)/(x(1+log)^(n))`

`int1/(x^2(x^4+1)^(3/4))dx`

`intsqrt(1+sin 2x) dx`

`int(x^(2) +3)/(x^(2)+1)dx`

`int" cos"^(3)(3x +5) dx`

`int(2x^(3))/((x^(2)+1)^(2))dx`

Evaluate: `int1/(sin(x-a)sin(x-b)) dx`

Let `f(x)=(sin^(2)pix)/(1+pi^(x)).` Then, `int[f(x)+f(-x)]dx` is equal to

`int(x+1)(x+2)^7(x+3)dx` is equal toA. `((x+2)^(10))/(10)-((x+2)^(8))+C`B. `((x+1)^(2))/(2)-((x+2)^(8))/(8)-((x+3)^(2))/(2)+C`C. `((x+2)^(10))/(10)+C`D. `((x+1)^(2))/(2)+((x+2)^(8))/(8)+((x+3)^(2))/(2)+C`

` int(1)/(5+4 cos x)dx`

`(i)intsec^(7) x. sin x dx" "(ii) int(1)/(sinx. cos^(2) x)dx`

`int(1)/((asin x+b cos x)^(2))dx`

The value of `int(x^(2)+1)/(x^(4)-x^(2)+1)dx` isA. `tan^(-1)(2x^(2)-1)+C`B. `tan^(-1)(x^(2)+1)/(x)+C`C. `sin^(-1)(x-(1)/(x))+C`D. `tan^(-1)((x^(2)-1)/(x))+C`

Evaluate: (i) `int1/(x^2)cos^2(1/x) dx`(ii) `intsec^4xtanx dx`

`int" cos"^(4) " 2x dx "`

`int(1)/(sqrt(x^(2) -9))dx`

If `(d(f(x))/(dx)=(1)/(1+x^(2)) `then `(d)/(dx){f(x^(3))} `isA. `(3x)/(1+x^(3))`B. `(3x^(2))/(1+x^(6))`C. `(-6x^(5))/((1+x^(6))^(2))`D. `(-6x^(5))/(1+x^(6))`

`int ((x+2)/(x+4))^2 e^x dx` is equal toA. `e^(x)((x)/(x+4))+C`B. `e^(x)((x+2)/(x+4))+C`C. `e^(x)((x-2)/(x+4))+C`D. `((2xe^(x))/(x+4))+C`

`int(1)/(2+sin^(2) x)dx`

`int(1)/(4x^(2) +9) dx`

`int(2x-1)/(sqrt(x^(2)-x-1))dx`

`int(dx)/(cos x-sinx)` is equal toA. `(1)/(sqrt2)log|tan((x)/(2)-(pi)/(8))|+C`B. `(1)/(sqrt2)log|cot((x)/(2))|+C`C. `(1)/(sqrt2)log|tan((x)/(2)-(3pi)/(8))|+C`D. `(1)/(sqrt2)log|tan((x)/(2)+(3pi)/(8))|+C`

`int(x^2dx)/(4+x^2)`

`(i) int sin 2x. cos5x dx " "(ii) int(sin 4x)/(sin x) dx`

`int(x^(2) +1)/((x+1))dx`

`int cos 2x . cos 4x . cos 6x dx`

`int(sin^(6)x+cos^(6)x+3sin^(2)x cos^(2)x)dx` is equal toA. `x+C`B. `(3)/(2)sin2x+C`C. `-(3)/(2)cos 2x+C`D. `(1)/(3)sin 3x -cos 3x+C`

`int(1)/(sqrt(5-.(x^(2))/(4)))dx`

`int sin^(2//3) x cos^(3) x dx`

`intsec^(4) x dx`