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1.

Evaluate:`int1/(4cos^2x+9sin^2x)dx`

Answer» `int(1/cos^2x)/(4+(9sin^2x)/(cos^2x))dx`
Let`t=tanx`
`dt=secc^2xdx`
`intsec^2x/(4+9tan^2x)dx`
`intdt/(4+9t^2)`
`1/9*int(dt)/(4/9+t^2)`
`1/9*3/2*tan^(-1)(t/(2/3))+c`
`1/6*tan^(-1)((3t)/2)+c`
`1/6*tan^(-1)((3tanx)/2)+c`.
Discussion
2.

If `int sqrt(cos^3x/sin^11x) dx = -2(A tan^(9//2)x + B tan^(5//2) x) + C,` then find A and B.A. `A=(1)/(9),B=-(1)/(5)`B. `A=(1)/(9),B=(1)/(5)`C. `A=-(1)/(9),B=(1)/(5)`D. none of these

Answer» Correct Answer - b
We observe that the sum of the exponents of cos x and sin x is `((3)/(4)-(11)/(2))=-4=a` negative even integer.
So , we divide numberator and denominatior by `cos^(4)x`
`becauseI=intsqrt((cos^(3)x)/(sin^(11)x))dx=int(cos^(3//2)x)/(sin^(11//2)x)dx`
`rArrI=int(cos^(3//2)x xxcos^(4)x)/(sin^(11//2)x)xx(1)/(cos^(4)x)dx`
`rArrI=int(1)/(tan^(11//2)x)(1+tan^(2)x)d(tanx)`
`rArrI=int(tan^(-11//2)x+tan^(-7//2)x)d (tanx)`
`rArr=-(2)/(9)tan^(-9//2)x-(2)/(5)tan^(-5//2)x+C`
`rArr I=-{(1)/(9)tan^(-9//2)x+(1)/(5)tan^(-5//2)x}+C`
Hence , `A=(1)/(9)andB=(1)/(5)`
Discussion
3.

`int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx` is equal toA. `log_(e)(tan^(2)x+sqrt(1+tan^(4)x))+C`B. `(1)/(2)log_(e)(tan^(2)x+sqrt(1+tan^(4)x))+C`C. `(1)/(4)log(tan^(2)x+sqrt(1+tan^(4)x))+C`D. none of these

Answer» Correct Answer - b
Let `I=int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx`
`rArr I=int(tan xsec^(2)x)/(sqrt(tan^(4)x+1))dx=-(1)/(2)int(1)/(sqrt((tan^(2)x)^(2)+1))d(tan^(2)x)`
`rArrI=(1)/(2)log_(e){tan^(2)x+sqrt(1+tan^(4)x)}+C`
Discussion
4.

The value of `int(1-x^7)/(x(1+x^7))dx` is equal toA. `a=1,b=(2)/(7)`B. `a=-1,b=(2)/(7)`C. `a=1,b=-(2)/(7)`D. `a=-1,b=-(2)/(7)`

Answer» Correct Answer - c
We have , `int(1-x^(7))/(x(1+x^(7)))dx=aIn |x| +bIn|x^(7)+1|+C`
Differentiating both sides W.r.t to , x, to x, we get
`(1-x^(7))/(x(1+x^(7)))=(a)/(x)+7b(x^(6))/(x^(7)+1)`
`rArr1-x^(7)=a(1+x^(7))+7bx^(7)rArra=1,b=-(2)/(7)`
Discussion
5.

`int(x-x^5)^(1/5) / x^6 dx`A. `(5)/(24)((1)/(x^(4))-1)^(6//5)+C`B. `(5)/(24)(1-(1)/(x^(4)))^(6//5)+C`C. `-(5)/(24)(1-(1)/(x^(4)))^(6//5)+C`D. none of these

Answer» Correct Answer - c
Let `I=int((x-x^(5))^(1//5))/(x^(6))dx=int((1)/(x^(4))-1)^(1//5)(1)/(x^(5))dx`
`rArrI=-(1)/(4)int((1)/(x^(4))-1)^(1//5)((-4)/(x^(5)))dx=-((1)/(x^())-1)^(1//5)d((1)/(x^(4))-1)`
`rArr I=-(5)/(24)((1)/(x^(4))-1)^(6//5)+C`
Discussion
6.

Evaluate:`(sin^3x dx)/((cos^4x+3cos^2x+1)tan^(-1)(secx+cosx)`A. `tan^(-1)(secx+cosx)+C`B. `log_(e)|tan^(-1)(secx+cosx)|+C`C. `(1)/((secx+cosx)^(2))+C`D. none of these

Answer» Correct Answer - b
Let `I=int(sin^(3)x)/((cos^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx`
`rArrI=int((sin^(3)x)/(cos^(2)x))/((cos^(2)x+3+sec^(2)x)tan^(-1)(secx+cosx))dx`
`I=int(1)/(1+(secx+cosx)^(2))xx(sinx(1-cos^(2)x))/(cos^(2)x)xx(1)/(tan^(-1)(secx+cosx))dx`
`rArrI=int(1)/(tan^(-1)(secx+cosx))xx(1)/(1+(secx+cosx)^(2))xx(tanxsec x-sinx)dx`
`rArrI=int(1)/(tan^(-1)(secx+cosx))d{tan^(-1)(secx+cosx)}`
`rArrI=log|tan^(-1)(secx+cosx)|+C`
Discussion
7.

`int(3 sinx+ 2 cos x)/(3 cosx+2 sin x)dx= ax+b log |3cos x + 2 sin x| +C`, then (a ,b)A. `a=(5)/(13), b =-(12)/(13)`B. `a=(12)/(13),b =-(5)/(13)`C. `a=(12)/(13),b=(5)/(13)`D. `a=(-12)/(5),b=(-5)/(13)`

Answer» Correct Answer - b
Let `I= int(3 sin x+2 cos x)/(3 cos x+2sinx)dx`
Let `3 sin x +2 cos x= lambda (d)/(dx)(3cosx +2sinx)+mu(3 cos x +2 sinx)`
`rArr 3 sin x +2 cos x = lambda(-3 sin x +2 cos x )+mu (3 cos x+2 sin x)`
Comparing the coefficients sinx and cos x on both sides , we get
`-3lambda_2mu = 3 and 2 lambda +3 mu =2`
` rArr lambda =-5 //13 and = 12//13`
`:. I= int(-((5)/(13))(-3sin x +2 cos x)+((12)/(13))(3 cos x +2 sin x))/(3 cos x+ 2 sin x)dx`
`rArr I=(12)/(13)int1*dx - (5)/(13) int (1)/(3 cos x +2sin x)d(3cos x +2 sin x)`
`rArr I=(12)/(13) x-(5)/(13)log |3 cos x+2 sin x| +C`
Hence , a `=(12)/(13) and b = (-5)/(13)`
Discussion
8.

Let F(x) be an indefinite integral of `sin^(2)x` Statement I The function F(x) satisfies `F(x+pi)=F(x)` for all real x. Because Statement II `sin^(2)(x+pi)=sin^(2)x,` for all real x.A. Statement - 1 True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.B. Statement - 1 is True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.C. Statement - 1 True ,Statement - 2 is False.D. Statement - 1 is False , Statement - 2 is True.

Answer» It is given that `F(x)=intsin^(2)x dx=(1)/(2)int(1-cos2x)dx=(1)/(2)(x-(sin2x)/(2))+C`
`thereforeF(pi+x)-F(x)=(1)/(2)[{(pi+x)-(1)/(2)sin2(pi+x)}-{x-(1)/(2)sin2x}]`
`=(1)/(2)(pi-(1)/(2)sin2x+(1)/(2)sin2x)=(pi)/(2)` for all x thus,F `(x+pi)` = F (x) for all x is not true.
So , statement -2 is not true.
Clearly , statement -2 is true as `sin^(2)` a is periodic with period `pi`.
Discussion
9.

If `intsec^(4//3)c "cosec"^(8//3)x dx =a(tanx)^(-5//3)+b(tanx)^(1//3)+C`, then 5a +b=A. 3B. `-3`C. 0D. `-1`

Answer» Correct Answer - c
We have , `Iint sec^(4//3)x " cosec"^(8//3) x dx`
`rArr I=intcos ^(-4//3)x sin^(-8//3)x dx = int(1)/(cos^(4//3)x sin^(8//3)x)dx` The sum of the exponent of sin x and cos x is -4 ,an even integer . So , we divide both numerator and denominator by ` cos^(4)` x.
`:. I= int (sec^(4)x)/(tan^(8//3)x)dx=int(1+tan^(2)x)(tanx)^(-8//3)d (tanx)`
`rArr I=int {(tanx)^(-8//3)+(tanx)^(-2//3)}d (tanx)`
`rArr I=-(3)/(5)(tanx)^(-5//3)+3(tanx)^(1//3)+C`
`:. a=-(3)/(5)and b =3 rArr 5a +b=0`
Discussion
10.

The value of `int(sinx +cosx)/(3+sin2x)dx` , isA. `(1)/(4)log((2+sinx - cosx)/(2-sinx+cosx))+C`B. `(1)/(2)log((2+sinx)/(2-sinx))+C`C. `(1)/(4)log((1+sinx)/(1-sinx))+C`D. none of these

Answer» Correct Answer - a
We have, `I=int(sinx +cosx)/(3+sin2x)dx`
`rArr I=int(1)/(4-(1-sin2x))d (sinx-cosx)`
`rArrI-int(1)/(2^(2)-(sinx-cosx)^(2))d(sinx - cosx)`
`rArr I=(1)/(4)log((2+sin x- cosx)/(2- sin x + cos x))+C`
Discussion
11.

If `intsqrt((x)/(a^(3)-x^(3)))dx=msin^(-1)((x)/(a))^(n)+C`, thenA. m = nB. m =-nC. `m=1//n`D. `m=-1//n`

Answer» Correct Answer - c
We have , `I=intsqrt((x)/(a^(3)-x^(3)))dx = int(sqrt(x))/(sqrt((a^(3//2))^(2)-(x^(3//2))^(2))dx`
`rArr I=(2)/(3)int(1)/(sqrt((a^(3//2))^(2)-(x^(3//2))^(2)))d (x^(3//2))`
`rArrI=(2)/(3)sin^(-1)((x)/(a))^(3//2)+C`
`:. m = (2)/(3)andn=(3)/(2) rArr m = (1)/(n)`
Discussion
12.

`int(1)/((x-1)sqrt(x^(2)-1))dx` equalsA. `-sqrt((x-1)/(x+1))+C`B. `sqrt((x-1)/(x+1))+C`C. `sqrt((x+1)/(x-1))+C`D. `-sqrt((x+1)/(x-1))+C`

Answer» Correct Answer - d
Let `I=int(1)/((x-1)sqrt(x^(2))-1)dx=int(1)/((x-1)^(3//2)sqrt(x+1))dx`
`rArrI=int((x+1)/(x-1))^(3//2)(1)/((x+1)^(2))dx=(1)/(2)int((x-1)/(x+1))^(-3//2)d((x-1)/(x+1))`
`rArrI=-((x-1)/(x+1))^(-1//2)+C=-sqrt((x+1)/(x-1))+C`
Discussion
13.

If `I_n=int( lnx)^n dx` then `I_n+nI_(n-1)`A. `(xlogx)^(n)`B. `x(logx)^(n)`C. `n(logx)^(n)`D. `(logx)^(n-1)`

Answer» Correct Answer - b
Let `I_(n)=intunderset(I)((logx)^(n))*underset(II)(1)dx`
`rArrI_(n)=x(logx)^(n)=intxn(logx)^(n-1)(1)/(x)dx`
`rArr I_(n)=x(logx)^(n)-nI_(n-1)`
`rArrI_(n)+nI_(n-1)=x(logx)^(n)`
Discussion
14.

Integration of `(1)/(sqrt(x^(2)=9))` with respect to `(x^(2)+1)` is equal toA. `sqrt(x^(2)+9)+C`B. `-(1)/(sqrt(x^(2)+9))+C`C. `2sqrt(x^(2)+9)+C`D. none of these

Answer» Correct Answer - c
Discussion
15.

`"If"int(dx)/(x^(3)(1+x^(6))^(23))=xf(x)(1+x^(6))^(1/3)+C` where, C is a constant of integration, then the function f(x) is equal toA. `-(1)/(2)`B. `-(1)/(6)`C. `-(6)/(x)`D. `-(x)/(2)`

Answer» Correct Answer - a
Let `=int(1)/(x^(3)(1+x^(6))^(2//3))dx` . Then
`I=int(1)/(7(1+(1)/(x^(6)))^(2//3))dx=-(1)/(6)int(1+(1)/(x^(6)))^(-2//3)d(1+(1)/(x^(6)))`
`rArrI=-(1)/(2)(1+(1)/(x^(6)))^(1//3)+C`
`rArrf(x)(1+x^(-6))^(1//3)+C=-(1)/(2)(1+x^(-6))^(1//3)+C`
`rArrf(x)=-(1)/(2)`.
Discussion
16.

`int(x^(3))/((1+x^(2))^(1//3))dx` is equal toA. `(20)/(3)(1+x^(2))^(2//3)(2x^(2)-3)+C`B. `(3)/(20)(1+x^(2))^(2//3)(2x^(2)-3)+C`C. `(3)/(20)(1+x^(2))^(2//3)(2x^(2)+3)+C`D. none of these

Answer» Correct Answer - b
Discussion
17.

What is `int (x^(4) -1)/(x^(2) sqrt(x^(4) + x^(2) + 1)) dx` equal to ?A. `(x)/(sqrt(x^(4)+x^(2)+1))+C`B. `(sqrt(x^(4)+x^(2)+1))/(x)+C`C. `(2x)/(sqrt(x^(4)+x^(2)+1))+C`D. `(sqrt(x^(4)+x^(2)+1))/(2x)+C`

Answer» Correct Answer - b
Discussion
18.

`int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2)))dx`A. In `|(sqrt(2xe^(sinx)+1)-1)/(sqrt(2xe^(sinx)+1)+1)|+C`B. In `|(sqrt(2xe^(sinx)-1)-1)/(sqrt(2xe^(sinx)-1)+1)|+C`C. In `|(sqrt(2xe^(sinx)-1)+1)/(sqrt(2xe^(sinx)-1)-1)|+C`D. In `|(sqrt(2xe^(sinx)+1)+1)/(sqrt(2xe^(sinx)-1)+1)|+C`

Answer» Correct Answer - a
Discussion
19.

`int(x^(4)+1)/(x^(6)+1)dx` is equal toA. `tan^(-1)x+(1)/(3)tan^(-1)x^(3)+C`B. `tan^(-1)x-(1)/(3)tan^(-1)x^(3)+C`C. `-tan^(-1)x-(1)/(3)tan^(-1)x^(3)+C`D. none of these

Answer» Correct Answer - a
`I=int(x^(4)+1)/(x^(6)+1)dx=int((x^(4)-x^(2)+1)+x^(2))/(x^(6)+1)dx`
`rArr I=int(x^(4)-x^(2)+1)/((x^(2))+1)dx+int(x^(2))/(x^(6)+1)dx`
`rArr I= int(1)/(x^(2)+1)dx+(1)/(3)int(1)/((x^(3))^(2)+1)d(x^(3))`
`rArr I=tan^(-1)x+(1)/(3)tan^(-1)(x^(3))+C`
Discussion
20.

`inte^(x)(1-cotx+cot^(2)x)dx=`A. `e^(x)cotx+C`B. `-e^(x)cotx+C`C. `e^(x)"cosec "x+C`D. `-e^(x)"cosec " x+C`

Answer» Correct Answer - b
We have , `I=inte^(x)(1-cotx+cot^(2)x)dx=inte^(x)(1+cot^(2)x-cotx)dx`
`rArr I=inte^(x)(-cotx + " cosec "^(2)x)dx=-e^(x)cotx+C`
Discussion
21.

`int{(logx-1)/(1+(logx)^(2))}^(2)` dx is equal toA. `(x)/((logx)^(2)+1)+C`B. `(xe^(x))/(1+x^(2))+C`C. `(x)/(1+x^(2))+C`D. `(logx)/((logx)^(2)+1)+C`

Answer» Correct Answer - a
We have , `I=int{(logx-1)/(1+(logx)^(2))}^(2)dx`
`rArr I=inte^(t)((t-1)^(1))/((t^(2)+1)^(2))dt` , where t = log x
`rArrI=inte^(t)(t^(2)+1-2t)/((t^(2)+1)^(2))dt`
`rArr I=inte^(t){(1)/(underset(f)(t^(2)+1))+(-2t)/(underset(f)((t^(2)+1)^(2)))}dt`
`rArrI=(e^(t))/(t^(2)+1)+C=(x)/((logx)^(2)+1)+C`
Discussion
22.

`int(1+x-x^(- 1))e^(x+x^(- 1))dx=`A. `(x+1)e^(x+x^(-1))+C`B. `(x-1)e^(x+x^(-1))+C`C. `-xe^(x+x^(-1))+C`D. `xe^(x+x^(-1))+C`

Answer» Correct Answer - d
Let `I=int(1+x-x^(-1))e^(x+x^(-1))dx`
`rArrI=inte^(x+x^(-1))dx+intx(1-(1)/(x^(2)))e^(x+x^(-1))dx`
`rArr I=inte^(x+x^(-1))dx+intunderset(I)(x)e^(x+x^(-1))d(x+x^(-1))`
`rArr I=inte^(x+x^(-1))dx+xe^(x+x^(-1))-inte^(x+x^(-1))dx`
`rArrI=xe^(x+x^(-1))+C`
Discussion
23.

If `int (e^x-1)/(e^x+1)dx=f(x)+C,` then f(x) is equal toA. `2log(e^(x)+1)+C`B. `log(e^(2x)-1)+C`C. `2log(e^(x)+1)-x+C`D. `log(e^(2x)+1)+C`

Answer» Correct Answer - c
We have , `int(e^(x)-1)/(e^(x)+1)dx`
`rArr = int(e^(x))/(e^(x)+1)dx-int(1)/(e^(x)+1)dx`
`rArr = int(1)/(e^(x)+1)d(e^(x)+1)+int(1)/(e^(-x)+1)d(e^(-x)+1)`
`rArr=log(e^(x)+1)+log(e^(-x)+1)+C`
`rArr=log(e^(x)+1)+log((e^(x)+1)/(e^(x)))+C`
`rArr=log(e^(x)+1)^(2)-loge^(x)+C=2log(e^(x)+1)-x+C`
Discussion
24.

The value of `intxlogx(logx-1)dx` is equal toA. `2(xlogx-x)^(2)+C`B. `(1)/(2)(xlogx-x)^(2)+C`C. `(x logx)^(2)+C`D. `(1)/(2)(x logx)^(3)+C`

Answer» Correct Answer - b
Let `I=intx logx (logx-1)dx=intlogx(xlogx-x)dx`
`rArrI=int(xlogx-x)d (xlogx-x)=((xlogx-x)^(2))/(2)+C`
Discussion
25.

If `l^r(x)` means `logloglog.......x` being repeated `r` times, then `int [ (x l(x) l^2(x) l^3(x) .... l^r (x)]^(-1) dx` is equal to :A. `l^(r+1)(x)+C`B. `(l^(r+1)(x))/(r+1)+C`C. `l^(r)(x)+C`D. none of these

Answer» Correct Answer - a
Discussion
26.

The value of the integral `int(1+x^(2))/(1+x^(4))dx` is equal toA. `tan^(-1)x^(2)+C`B. `(1)/(sqrt(2))tan^(-1)((x^(2)-1)/(sqrt(2)x))`C. `(1)/(2sqrt(2))log((x^(2)+sqrt(2)x+1)/(x^(2)-sqrt(2)x+1))+C`D. none of these

Answer» Correct Answer - b
Discussion
27.

`I=int(1)/((a^(2)-b^(2)x^(2))^(3//2))dx` is equal toA. `(x)/(sqrt(a^(2)-b^(2)x^(2)))+C`B. `(x)/(a^(2)sqrt(a^(2)-b^(2)x^(2)))+C`C. `(ax)/(sqrt(a^(2)-b^(2)x^(2)))+C`D. none of these

Answer» Correct Answer - b
We have `I=int(1)/((a^(2)-b^(2)x^(2))^(3//2))=(-1)/(2a^(2))int(-2a^(2))/(x^(3)((a^(2))/(x^(2))-b^(2))^(3//2))dx`
`rArr I=(-1)/(2a^(2))int((a^(2))/(x^(2))-b^(2))^(-3//2)d((a^(2))/(x^(2))-b^(2))`
`rArrI=(1)/(a^(2))((a^(2))/(x^(2))-b^(2))^(-1//2)+C(x)/(a^(2)(a^(2)-b^(2)x^(2))^(1//2))+C`
Discussion
28.

`int(e^x)/((1+e^x)(2+e^x))dx`A. `log((e^(x)+1)/(e^(x)+2))+C`B. `log((e^(x)+2)/(e^(x)+1))+C`C. `(e^(x)+1)/(e^(x)+2)+C`D. `(e^(x)+2)/(e^(x)+1)+C`

Answer» Correct Answer - b
Discussion
29.

The value of `int(1)/(x+sqrt(x-1))dx`, isA. `log(x+sqrt(x-1))+sin^(-1)sqrt((x-1)/(x))+C`B. `log(x+sqrt(x-1))+C`C. `log(x+sqrt(x-1))-(2)/(3)tan^(-1)((2sqrtx-1+1)/(sqrt(3)))+C`D. none of these

Answer» Correct Answer - c
Let `x-1=t^(2)`. Then,
`int(1)/(x+sqrt(x-1))dx=int(t)/(t^(2)+t+1)dt=int((2t +1)-1)/(t^(2)+t+1)dt`
`=int(2t+1)/(t^(2)+t+1)dt =int(1)/(t+(t+(1)/(2))^(2)+((sqrt(3))/(2))^(2))dt`
`=log (t^(2)+t+1)-(2)/(sqrt(3))tan^(-1)((2t+1)/(sqrt(3)))+C`
`=log (x+sqrt(x-1))-(2)/(sqrt(3))tan^(-1)((2sqrt(x-)+1)/(sqrt(3)))+C`
Discussion
30.

If `int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx=(1)/(12)tan^(-1)(3 tanx)+C`, then the value of ab , isA. `sqrt(41)`B. `sqrt(40)`C. `sqrt(39)`D. `sqrt(38)`

Answer» Correct Answer - b
We have , `I= int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx`
`rArr I= int(sec^(2)x)/(b^(2)+a^(2)tan^(2)x)dx`
` rArr I=(1)/(a)int(1)/(b^(2)+(atanx)^(2))d(atanx)`
`=(1)/(ab) tan^(-1)((a)/(b)tanx)+C`
`:. ab = 12 and (a)/(b) = 3 rArr a^(2)= 36 rArr a = +-6`
`:. ab = 12 rArr b =+-2`.
Thus , we have
`a sin x +b cos x = +- (6 sin x+2cosx)` We know that
`-sqrt(a^(2)+b^(2))le a sin x +b cos x le sqrt(a^(2)+b^(2))` for all x
` :. - sqrt(40)le +- 6 sin x +- 2 cos x le sqrt(40)` for allx.
Discussion
31.

`int(1)/(1+3 sin ^(2)x)dx` is equal toA. `(1)/(3)tan^(-1)(3 tan^(2)x)+C`B. `(1)/(2)tan^(-1)(2tan x)+C`C. `tan^(-1) (tanx)+C`D. none of these

Answer» Correct Answer - b
We have , `I=int(1)/(1+3 sin^(2)x)dx = int(sec^(2)x)/(sec^(2)x+3tan^(2)x)dx`
`rArr I=(1)/(2)int(1)/((2 tan x)^(2)+1^(2))d (2 tanx)`
`rArr I=(1)/(2)tan^(-1)(2tanx)+C`
Discussion
32.

The integral `int1/((1+x^2)sqrt(1-x^2))dx` is equal toA. `(1)/(2)tan^(-1)((sqrt(2)x)/(sqrt(1-x^(2))))`B. `(1)/(sqrt(2))tan^(-1)((sqrt(2)x)/(sqrt(1+x^(2))))`C. `(1)/(sqrt(2))tan^(-1)((sqrt(2)x)/(sqrt(1-x^(2))))`D. none of these

Answer» Correct Answer - c
Discussion
33.

`int(1)/(7+5 cos x)dx=`A. `(1)/(sqrt(6))tan^(-1)((1)/(sqrt(6))"tan" (x)/(2))+C`.B. `(1)/(sqrt(3))tan^(-1)((1)/(sqrt(3))"tan" (x)/(2))+C`C. `(1)/(4)tan^(-1)((x)/(2))+C`D. `(1)/(7)tan^(-1)("tan" (x)/(2))+C`

Answer» Correct Answer - a
We have , `I= int(1)/(7+5 cos x)dx= int(1+tan ^(2)x//2)/(7(1+"tan"^(2)(x)/(2))+5(1- " tan" ^(2)(x)/(2)))dx`
`rArr I= int ("sec"^(2)(x)/(2))/(12+2"tan"^(2)(x)/(2))dx=int((1)/(2)"sec"^(2)(x)/(2))/((sqrt(6))^(2)("tan"(x)/(2))^(2))dx`
`rArr I= int(1)/((sqrt(6))^(2)+("tan"(x)/(2))^(2))d ("tan"(x)/(2))=(1)/(sqrt(6))tan^(-1)((1)/(sqrt(6))"tan"(x)/(2))+C`
Discussion
34.

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

Answer» Correct Answer - d
We have , `I= int(1)/(cos x - sin x )dx = (1)/(sqrt(2))int(1)/(cos x " cos" (pi)/(4)- sin x " sin" (pi)/(4))dx`
`rarr I= (1)/(sqrt(2))int(1)/(cos(x+(pi)/(4)))dx= (1)/(sqrt(2))intsec(x+(pi)/(4))dx`
`rArr I= (1)/(sqrt(2))log|tan ((pi)/(4)+(x)/(2)+(pi)/(8))|+C= (1)/(sqrt(2))log|tan ((x)/(2)+(3pi)/(8))|+C`
Discussion
35.

`int1/(cosx+sqrt(3)sinx)` dx equalsA. `logtan((pi)/(2)+(pi)/(12))+C`B. `logtan((x)/(2)-(pi)/(12))+C`C. `(1)/(2)log tan ((x)/(2)+(pi)/(12))+C`D. `(1)/(2)log tan((x)/(2)-(pi)/(12))+C`

Answer» Correct Answer - c
We have , `I=int(1)/(cosx +sqrt(3)sin x)dx`
`rArr I= (1)/(2)int(1)/((1)/(2)cosx+(sqrt(3))/(2)sinx)dx`
`rArrI=(1)/(2)int (1)/(cos x "cos" (pi)/(3)+sin x " sin" (pi)/(3))dx`
`rArrI=(1)/(2)int(1)/(cos(x-(pi)/(3)))dx = (1)/(2) int sec(x-(pi)/(3))dx`
`rArr I= (1)/(2)log tan ((pi)/(4)+(pi)/(2)-(pi)/(6))+C = (1)/(2) log tan((x)/(2)+(pi)/(12))+C`
Discussion
36.

`int sqrt((cosx-cos^3x)/(1-cos^3x))dx` is equal toA. `(2)/(3)sin^(-1)(cos^(3//2x))+C`B. `(3)/(2)sin^(-1)(cos^(3//2x))+C`C. `(2)/(3)cos^(-1)(cos^(3//2)x)+C`D. none of these

Answer» Correct Answer - c
Discussion
37.

`int(1)/(sin(x-a)cos(x-b))dx` is equal toA. `(1)/(sin(a-b))log|(sin(x-a))/(cos(x-b))|+C`B. `(1)/(cos(a-b))log|(sin(x-a))/(cos(x-b))|+C`C. `(1)/(sin(a+b))log|(sin(x-a))/(cos(x-b))|+C`D. `(1)/(cos(a+b))log|(sin(x-a))/(cos(x-b))|+C`

Answer» Correct Answer - b
Discussion
38.

`int(1)/(x(1+root(3)(x))^(2))dx`is equal toA. `3{"log"((x^(1//3))/(1+x^(1//3)))+(1)/(1+root(3)(x))}+C`B. `3{"log"((x^(1//3))/(1+x^(1//3)))+(1)/(1+x^(1//3))}+C`C. `3{"log"((x^(1//3))/(1+x^(1//3)))-(1)/(1+x^(1//3))}+C`D. none of these

Answer» Correct Answer - a
Let `I=int(1)/(x(1+root(3)(x))^(2))dx=int(1)/(t^(3)(1+t)^(2))3t^(2)dt`
`rArr I=3int(1)/(t(t+1)^(2))dt=3 int{(1)/(t)-(1)/(t+1)-(1)/((t+1)^(2))}dt`
`rArr I=3 { log_(e)t-log(t+1)+(1)/(t+1)}+C`
`rArrI=3 {log_(e)((t)/(t+1))+(1)/(t+1)}+C`
`rArrI=3{log_(e)(x^(1//3)/(1+x^(1//3)))+(1)/(x^(1//3)+1)}+C`.
Discussion
39.

Let `f(x)=int x^2/((1+x^2)(1+sqrt(1+x^2)))dx` and `f(0)=0` then `f(1)` isA. `log_(e)(1+sqrt(2))`B. `log_(e)(1+sqrt(2))-(pi)/(4)`C. `log_(e)(1+sqrt(2))+(pi)/(4)`D. none of these

Answer» Correct Answer - b
We have , `f(x) = int(x^(2))/((x^(2)+1){1+sqrt(1+x^(2))})dx`
`rArrf(x) = int((sqrt(1+x^(2))-1))/((x^(2)+1))dx=int(1)/(sqrt(x^(2)+1))dx=int(1)/(x^(2)+1)dx`
`rArrf(x) = log_(e)(x+sqrt(x^(2)+1))-tan^(-1)x+C`
Now , `f(0) =0= rArr0=log_(e)1-tan^(-1)0+CrArrC=0`
`thereforef(x) =log_(e)(x+sqrt(x^(2)+1))-tan^(-1)x`
`rArr f(1)=log_(e)(sqrt(2)+1)-(pi)/(4)`
Discussion
40.

If `int(1)/((x+1)(x-2))dx=Alog_(e)(x+1)+Blog_(e)(x-2)+C` , then A + B = ?A. A+B=0B. A - B = 0C. AB =1D. AB =-1

Answer» Correct Answer - a
Let `I=int(1)/((x+1)(x-2))dx=int(1)/(3)((1)/(x-2)-(1)/(x+1))`dx
`rArrI=(1)/(3)log_(e)(x-2)-(1)/(3)log_(e)(x+1)+C`
`thereforeA=-(1)/(3)andB=(1)/(3)rArrA+B=0`
Discussion
41.

Evaluate:`int(sin2x)/(sin5xsin3x)dx`

Answer» `(sin2x)/(sin5xsin3x)=(sin(5x-3x))/(sin5xsin3x)`
`=(sin5xcos3x-cos5xsin3x)/(sin5xsin3x)`
`=(sian5xcos3x)/(sin5xsin3x)-(cos5xsin3x)/(sin5xsin3x)`
`=cot(3x)-cot(5x)`
`int(sin2x)/(sin5xsin3x)dx=int[cot3x-cot5x]dx`
`=1/3log|sin3x|-1/5log|sin5x|+c`.
Discussion
42.

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

Answer» Correct Answer - d
Let `1+x^(2)=t^(2)`. then x dx = t dt `thereforeI=int(x)/(sqrt(1+x^(2)+sqrt((1+x^(2))^(3))))dx`
`rArr I=int(t)/(sqrt(t^(2)+t^(3)))dt=int(1)/(sqrt(1+t))dt=2sqrt(1+t)+C`
`rArr I=2sqrt(1+sqrt(1+x^(2)))+C`
Discussion
43.

If `int f(x)dx=psi(x)`, then `int x^5f(x^3)dx`A. `(1)/(3)x^(3){x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx}+C`B. `(1)/(3)x^(3)phi(x^(3))-3intx^(3)phi(x^(3))dx+C`C. `(1)/(3)x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx+C`D. `(1)/(3){x^(3)phi(x^(3))-intx^(3)phi(x^(3))dx}+C`

Answer» Correct Answer - c
Let `x^(3)=t`. then , `d(x^(3))=3dtrArr3x^(2)dx=dt`
`thereforeintx^(5)f(x^(3))dx`
`=(1)/(3)intx^(3)f(x^(3))(3x^(2))dx`
`=(1)/(3)intunderset(I)(t)f(t)underset(II)dt`
`=(1)/(3){tphi(t)-intphi(t)dt}+C` " " `[becauseintf(t)dt=phi(t)]`
`=(1)/(3){x^(3)phi(x^(3))-3intphi(x^(3))x^(2)dx}+C`
`=(1)/(3)x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx+C`
Discussion
44.

The integral `int(1+x-1/x)e^(x+1/x)dx` is equal toA. `(x+1)e^(x+(1)/(x))+C`B. `-xe^(x+(1)/(x))+C`C. `(x-1)e^(x+(1)/(x))+C`D. `xe^(x+(1)/(x))+C`

Answer» Correct Answer - d
We have , `int(1+x-(1)/(x))e^(x+(1)/(x))dx`
`=inte^(x+(1)/(x))dx+intunderset(I)(x) e^(x+(1)/(x))(1-(1)/(x^(2)))dx`
`=inte^(x+(1)/(x))dx+xe^(x+(1)/(x))-inte^(x+(1)/(x))dx+C=xe^(x+(1)/(x))+C`
Discussion
45.

`int(x^(7m)+x^(2m)+x^m)(2x^(6m)+7x^m+14)^(1/m)dx`A. `(7x^(7m)+2x^(2m)+14x^(m))^((m+1)/(m))/(14(m+1))+C`B. `(2x^(7m)+14x^(2m)+7x^(m))^((m+1)/(m))/(14(m+1))+C`C. `(2x^(7m)+7x^(2m)+14x^(m))^((m+1)/(m))/(14(m+1))+C`D. `(7x^(7m)+2x^(2m)+x^(m))^((m+1)/(m))/(14(m+1))+C`

Answer» Correct Answer - c
Let `I=int(x^(7m)+x^(2m)+x^(m))(2x^(6m)+7x^(m)+14)^(1//m)dx`. then ,
`I=int(x^(7m-1)+x^(2m-1)+x^(m-1))(2x^(7m)+7x^(2m)+14x^(m))^(1//m)dx` Let `2x^(7m)+7x^(2m)+14x^(m)=t` . then,
`14m(x^(7m-1)+x^(2m-1)+x^(m-1))dx =dt`
`I=(1)/(14m)int^(t^(1//m))dt=(1)/(14m)(t^(1/(m)+1))/(((1)/(m)+1))+C`
`rArrI=(1)/(14(m+1))(2x^(7m)+7x^(2m)+14x^(m))^((m+1)/(m))+C`
Discussion
46.

if `int(1-5sin^2x)/(cos^5xsin^2x)dx=f(x)/(cos^5x)+c` then `f(x)`A. `-cotx`B. `-"cosec "x`C. `"cosec "x`D. cot x

Answer» Correct Answer - d
Let `I=int(1-5 sin^(2)x)/(cos^(5)x sin^(2)x)dx` . Then,
`I=int((cos^(2)x+sin^(2)x)-5sin^(2)x)/(cos^(5)x sin ^(2)x)dx`
`rArr I=int(cos^(2)x-4sin^(2)x)/(cos^(5)xsin^(2)x)dx=int(cos^(5)x-4sin^(2)xcos^(3)x)/(cos^(8) xsin^(2)x)dx`
`rArrI=int(d)/(dx)((1)/(cos^(4)xsinx))=(1)/(cos^(4)xsinx)+C`
`rArrI=(cotx)/(cos^(5)x)+C`
`rArr(f(x))/(cos^(5)x)+C=(cotx)/(cos^(5)x)+C`
`rArr f(x) = cotx`.
Discussion
47.

`int(1+x+sqrt(x+x^2))/(sqrt(x)+sqrt(1+x))dxi se q u a lto``1/2sqrt(1+x)C`(b) `2/3(1+x)^(x/2)+C``sqrt(1+x)+c`(d) `3/2(1+x)^(3/2)+C`A. `(1)/(2)sqrt(1+x)+C`B. `(2)/(3)(1+x)^(3//2)+C`C. `sqrt(1+x)+C`D. `2(1+x)^(3//2)+C`

Answer» Correct Answer - b
Discussion
48.

The integral `int (2x-3)/(x^2+x+1)^2 .dx` is equal toA. `-(8x+7)/(x^(2)+x+1)-(16)/(3sqrt(3))tan^(-1)((2x+1)/(3))+C`B. `-(1)/(x^(2)+x+1)-(4)/(3)tan^(-1)(4x+3)+C`C. `(1)/(2(x^(2)+x+1))-((2x+1)^(2))/((x^(2)+x+1)^(2))+C`D. `(1)/(4(x^(2)+x+1))+(2)/(3)tan^(-1)(2x+1)+C`

Answer» Correct Answer - a
Discussion
49.

`int(x+1)^(2)e^(x)dx` is equal toA. `xe^(x)+C`B. `x^(2)e^(x)+C`C. `(x+1)e^(x)+C`D. `(x^(2)+1)e^(x)+C`

Answer» Correct Answer - d
We have , `I=int(x+1)^(2)e^(x)dx=inte^(x)(x^(2)+2x)dx+inte^(x)dx`
`rArr I=e^(x)x^(2)+e^(x)+C=e^(x)(x^(2)+1)+C`
Discussion
50.

`inte^(x)(1+x)sec^(2)(xe^(x))dx=f (x)+` Constant , then f (x) is equal toA. `cos(xe^(x))`B. `sin(xe^(x))`C. `2tan^(-1)x`D. `tan(xe^(x))`

Answer» Correct Answer - d
We have, `inte^(x)(1+x)sec^(2)(xe^(x))dx`
`=intsec^(2)(xe^(x))d(xe^(x))=tan(xe^(x))+C`
`:. f(x) = tan (xe^(x))`
Discussion