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351.

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

Answer» Correct Answer - a
Discussion
352.

`int(6x-5)/(sqrt(3x^2-5x+1))dx`

Answer» `2sqrt(3x^(2)-5x+1)+C`
Discussion
353.

`int("sin"(5x)/(2))/("sin"(x)/(2))dx` is equal to (where, C is a constant of integration)A. `2x + "sin" x + 2 "sin" 2x + C`B. `x + 2"sin" x + 2 "sin" 2x + C`C. `x + 2"sin" x + "sin" 2x + C`D. `2x + "sin" x + "sin" 2x + C`

Answer» Correct Answer - C
Let `I=int("sin"(5x)/(2))/("sin"(x)/(2))dx = int("2 sin"(5x)/(2)"cos"(x)/(2))/("2 sin"(x)/(2)"cos"(x)/(2))dx` [multiplying by `"2 cos"(x)/(2)` in numerator and denominator]
`=int(sin3x + sin2x)/(sin x)dx`
`[therefore 2 sin A cos B = sin(A+B)+sin(A-B)and sin 2 A = 2 sin A cos A]`
`= int((3 sin x - 4 sin^(3) x)+2 sin x cos x)/(sin x)dx" "[therefore sin 3x = 3 sin x -4 sin^(3) x]`
`= int(3-4 sin^(2)x + 2 cos x)dx`
`=int[3-2(1-cos 2x)+2 cos x]dx" "[therefore 2 sin^(2)x=1 - cos 2x]`
`= int[3-2+2 cos 2x + 2 cos x]dx`
`=[1+2 cos 2x + 2 cos x]dx`
`=x+2 sin x + sin 2x + C`
Discussion
354.

If `int (x+1)/(sqrt(2x-1))dx = f(x)sqrt(2x-1)+C`, where C is a constant of integration, then f(x) is equal toA. `(2)/(3)(x+2)`B. `(1)/(3)(x+4)`C. `(2)/(3)(x-4)`D. `(1)/(3)(x+1)`

Answer» Correct Answer - B
....(i) We have, `int(x+1)/(sqrt(2x-1))dx = f(x)sqrt(2x-1)+C`
Let `I=int(x+1)/(sqrt(2x-1))dx`
Put `2x - 1 = t^(2) rArr 2dx = 2tdt rArr dx = tdt`
`I=int((t^(2)+1)/(2)+1)/(t)tdt = (1)/(2)int(t^(2)+3)dt" "[therefore 2x-1=t^(2) rArr x = (t^(2)+1)/(2)]`
`=(1)/(2)((t^(3))/(3)+3t)+C=(t)/(6)(t^(2)+9)+C`
`=(sqrt(2x-1))/(6)(2x-1+9)+C" "[therefore t = sqrt(2x-1)]`
`=(sqrt(2x-1))/(6)(2x+8)+C`
`=(x+4)/(3)sqrt(2x-1)+C`
On comparing it with Eq. (i), we get `f(x)=(x+4)/(3)`
Discussion
355.

Integrate: `int((5x^2-12)dx)/((x^2-6x+13)^2)`

Answer» `(13x-159)/(8(x^(2)-6x+13))+53/16tan^(-1)(x-3)/(2)+C`
Discussion
356.

`int(3x^(13)+2x^(11))/((2x^4+3x^2+1)^4)dx`A. `(x^(4))/(6(2x^(4)+3x^(2)+1)^(3))+C`B. `(x^(12))/(6(2x^(4)+3x^(2)+1)^(3))+C`C. `(x^(4))/((2x^(4)+3x^(2)+1)^(3))+C`D. `(x^(12))/((2x^(4)+3x^(2)+1)^(3))+C`

Answer» Correct Answer - B
Let `I=int(3x^(13)+2x^(11))/((2x^(4)+3x^(2)+1)^(4))dx = int((3)/(x^(3))+(2)/(x^(5)))/((2+(3)/(x^(2))+(1)/(x^(4)))^(4))dx` [on dividing numerator and denominator by `x^(16)`]
Now, put `2 + (3)/(x^(2))+(1)/(x^(4))=t`
`rArr ((-6)/(x^(3))-(4)/(x^(5)))dx = dt rArr ((3)/(x^(3))+(2)/(x^(5)))dx = - (dt)/(2)`
So, `I = int(-dt)/(2t^(4))= -(1)/(2)xx(t^(-4+1))/(-4+1)+C = (1)/(6t^(3))+C`
`= (1)/(6(2+(3)/(x^(2))+(1)/(x^(4)))^(3))+C" "[therefore t = 2 + (3)/(x^(2))+(1)/(x^(4))]`
`= (x^(12))/(6(2x^(4)+3x^(2)+1)^(3))+C`
Discussion
357.

Evaluate: `inte^(x)(x^(3)-x+2)/(x^(2)+1)^(2)`dx

Answer» `e^(x)(x+1)/(x^(2)+1)+C`
Discussion
358.

`inte^x((x^3-x+2)/((x^2+1)^2))dx`

Answer» `1/2e^(x)[(x^(2)-1)cosx+(x-1)^(2).sinx]+C`
Discussion
359.

`int x^(13/2) (1+x^5/2)^(1/2) dx = P(1+x^(5/2))^(7/2)+Q(1+x^(5/2))^(5/2) +R(1+x^(5/2))^(3/2)+C` then P, Q ,R areA. `P=4/35, Q=-8/25, R=4/15`B. `P=4/35, Q=8/25, R=4/15`C. `P=-4/35, Q=-8/25, R=4/15`D. `P=4/35, Q=-8/25,R=-4/15`

Answer» Correct Answer - A
Discussion
360.

The value of `int(1-x^7)/(x(1+x^7))dx` is equal toA. `ln|x|+2/7ln|1+x^(7)|+C`B. `ln|x|-2/7ln|1-x^(7)|+C`C. `ln|x|-2/7ln|1+x^(7)|+C`D. `ln|x|+2/7ln|1-x^(7)|+C`

Answer» Correct Answer - C
Discussion
361.

Evaluate: `cosx.e^(x).x^(2)dx`

Answer» `x,x^(2)+2xcosalpha+1`
Discussion
362.

`intdx/(sinx+secx)`A. `1/(2sqrt(3))log|(sqrt(3)+sinx-cosx)/(sqrt(3)-(sinx-cosx))|+tan^(-1)(sinx+cosx)+C`B. `1/(2sqrt(3))log|(sqrt(3)+sinx-cosx)/(sqrt(3)-(sinx-cosx))|+tan^(-1)(sinx-cosx)+C`C. `1/(2sqrt(3))log(sqrt(3)+sinx+cosx)/(sqrt(3)-(sinx-cosx))|+tan^(-1)sinx+cosx+C`D. `1/(2sqrt(3))log |(sqrt(3)+sinx-cosx)/(sqrt(3)-(sinx+cosx))|+tan^(-1)(sinx+cosx)+C`

Answer» Correct Answer - A
Discussion
363.

Evaluate `intx/((7x-10-x^2)^(3/2))dx`

Answer» `(2(7x-20))/(9sqrt(7x-10-x^(2))+C`
Discussion
364.

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

Answer» Correct Answer - C
Discussion
365.

`int x sinx sec^(3)x dx` is equal toA. `(1)/(2)[ sec^(2)x-tanx]+c`B. `(1)/(2)[x sec^(2)x-tanx]+c`C. `(1)/(2)[x sec^(2)x+tanx]+c`D. `(1)/(2)[ sec^(2)x+tanx]+c`

Answer» Correct Answer - B
`intx sinx sec^(3)x dx=intx sinx(1)/(cos^(3)x)dx`
`=int x tanx sec^(2)x dx `
`=x int secx(secx tanx)dx-int[int secx(secx tanx)dx]dx+C`
`=x(sec^(2)x)/(2)-int(sec^(2)x)/(2)dx+C`
`=x(sec^(2)x)/(2)-(tanx)/(2)+C`
Discussion
366.

Evaluate:`int(x+1)/(x(1+xe^x)^2)dx`

Answer» Correct Answer - `"log"|(xe^(x))/(1+xe^(x))|+(1)/(1+xe^(x))+c`
Let `I=int((x+1))/(x(1+xe^(x))^(2))dx=int(e^(x)(x+1))/(xe^(x)(1+xe^(x))^(2))dx`
Put `1 + xe^(x) = t rArr (e^(x)+xe^(x))dx=dt`
`therefore" "I=int(dt)/((t-1)t^(2))=int[(1)/(t-1)-(1)/(t)-(1)/(t^(2))]dt`
`=log|t-1|-log|t|+(1)/(t)+c`
`=log|(t-1)/(t)|+(1)/(t)+c`
`=log|(xe^(x))/(1+xe^(x))|+(1)/(1+xe^(x))+c`
Discussion
367.

If `inte^(x)/(x^(n))dx` and `-e^(x)/(k_(1)x^(n-1))+1/(k_(2)-1)I_(n-1)`, then `(k_(2)-k_(1))` is equal to:

Answer» Correct Answer - b
Discussion
368.

Evaluate: `int(x^(2)/(x^(4)+3x^(2)+9))dx =Atan^(-1)((x^2-3)/(3x))+B/sqrt(3)"ln"|(x^(2)-sqrt(3)x+3)/(x^(2)+sqrt(3)x+3)|+c`, Then find the value of `12(A+B)`.

Answer» Correct Answer - 5
Discussion
369.

The value of `inte^(tan^-1x) ((1+x+x^2)/(1+x^2))dx`A. `xe^(tan^(-1)x)+C`B. `x^(2)e^(tan^(-1)x)+C`C. `1/xe^(tan^(-1)x)+C`D. `xe^(cot^(-1)x+C`

Answer» Correct Answer - A
Discussion
370.

`int[(sqrt(x^2+1)(ln(x^2+1)-2lnx)))/(x^4) dx`

Answer» `(2(x^(2)+1)sqrt(x^(2)+1))/(9x^(3))[1-3/2"ln"(1+1/x^(2))]+C`
Discussion
371.

`"If " int x e^(x) cosx dx=ae^(x)(b(1-x)sinx+cx cosx)+d,` thenA. `a=1,b=1,c=-1`B. `a=(1)/(2),b=-1,c=1`C. `a=1,b=-1,c=1`D. `a=(1)/(2),b=1,c=-1`

Answer» Correct Answer - B
`I=int xe^(x)cosx dx `
`=xe^(x)sinx-int(xe^(x)+e^(x))sinx dx `
`=xe^(x)sinx-xe^(x)(-cosx)-int(xe^(x)+e^(x))cosx dx-int e^(x)sinx dx `
`=xe^(x)sinx+xe^(x)cosx-int xe^(x) cosx dx -int e^(x)(cosx+sinx)dx`
`"or " 2I=xe^(x)(sinx+cosx)-e^(x)sinx+d`
`=e^(x)((x-1)sinx+xcosx)+d`
`"or " I=(1)/(2)e^(x)((x-1)sinx+xcosx)+d`
`"or "a=(1)/(2),b=-1,c=1`
Discussion
372.

If `int(xe^(x))/(sqrt(1+e^(x)))dx=f(x)sqrt(1+e^(x))-2logg(x)+C`, thenA. `f(x)=x-1`B. `g(x)=(sqrt(1+e^(x))-1)/(sqrt(1+e^(x))+1`C. `g(x)=(sqrt(1+e^(x))+1)/(sqrt(1+e^(x))-1)`D. `f(x)=2(x-2)`

Answer» Correct Answer - B::D
`int(xe^(x))/(sqrt(1+e^(x)))dx`
`=x(sqrt(1+e^(x)))-2intsqrt(1+e^(x))dx`
`=2xsqrt(1+e^(x))-2int(2t^(2))/(t^(2)-1)dt" "("Putting t"=sqrt(1+e^(x)))`
`=2xsqrt(1+e^(x))-4(t+(1)/(2)ln.(t-1)/(t+1))+C`
`2xsqrt(1+e^(x))-4(sqrt(1+e^(x)))-2ln.(sqrt(1+e^(x))-1)/(sqrt(1+e^(x))+1)+C`
Discussion
373.

If `int x((ln(x+sqrt(1+x^2)))/sqrt(1+x^2)) dx=asqrt(1+x^2)ln(x+sqrt(1+x^2))+bx+c` thenA. `a=1,b=-1`B. `a=1,b=1`C. `a=-1,b=1`D. `a=-1,b=-1`

Answer» Correct Answer - A
`I=int x(In(x+sqrt(x^(2)+1)))/(sqrt(x^(2)+1))dx`
`"Let " t=sqrt(x^(2)+1)`
`"or " (dt)/(dx)=(x)/(sqrt(x^(2)+1))`
` :. I=int In(t+sqrt(t^(2)-1))dt `
`=In(t+sqrt(t^(2)-1))t-int(1+(t)/(sqrt(t^(2)-1)))/(t+sqrt(t^(2)-1))tdt`
`=t " In"(t+sqrt(t^(2)-1))-(1)/(2)int(2t)/(sqrt(t^(2)-1))dt `
`=t" In"(t+sqrt(t^(2)-1))-sqrt(t^(2)-1)+C`
`=sqrt(1+x^(2)) In (x+sqrt(1+x^(2)))-x+C`
`"or " a=1,b=-1`
Discussion
374.

If `I=inte^(-x)log(e^x+1)dx ,t h e nIe q u a l``a+(e^(-x)+1)log(e^x+1)+C``a+(e^x+1)log(e^x+1)+C``a-(e^(-x)+1)log(e^x+1)+C`none of theseA. `x+(e^(-x)+1)log(e^(x)+1)+C`B. `x+(e^(x)+1)log(e^(x)+1)+C`C. `x-(e^(-x)+1)log(e^(x)+1)+C`D. none of these

Answer» Correct Answer - C
`I=-e^(-x)log(e^(x)+1)+int(e^(-x)e^(x))/(e^(x)+1)dx `
`=-e^(-x)log(e^(x)+1)+int(e^(-x))/(e^(-x)+1)dx`
`=-e^(-x)log(e^(x)+1)-log(e^(-x)+1)+C`
`=-e^(-x)log(e^(x)+1)-log(1+e^(x))+x+C`
`=-(e^(-x)+1)log(e^(x)+1)+x+C`
Discussion
375.

`Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C ,`then`f(x)=1/2x^2`(b) `g(x)=logx``A=1`(d) none of theseA. `f(x)=(1)/(2)x^(2)`B. `g(x)=log x`C. `A=1`D. none of these

Answer» Correct Answer - D
` intx log(1+(1)/(x))dx `
`=int xlog(x+1)dx-int x logxdx `
` =(x^(2))/(2)log(x+1)-(1)/(2)int(x^(2))/(x+1)dx-(x^(2))/(2)logx+(1)/(2)int(x^(2))/(x)dx+C`
` =(x^(2))/(2)log(x+1)-(1)/(2)int(x-1+(1)/(x+1))dx-(x^(2))/(2)logx+(1)/(4)x^(2)+C`
`=(x^(2))/(2)log(x+1)-(x^(2))/(2)logx-(1)/(2)((x^(2))/(2)-x)-(1)/(2)log(x+1)+(1)/(4)x^(2)+C`
`=(x^(2))/(2)log(x+1)-(x^(2))/(2)logx-(1)/(2)log(x+1)+(1)/(2)x+C`
`"Hence, "f(x)=(x^(2))/(2)-(1)/(2),g(x)=-(1)/(2)logx, " and " A=(1)/(2).`
Discussion
376.

Evaluate:`int(ddx)/(sqrt(1+x^2)+sqrt((1+x^2)^3))`

Answer» Correct Answer - `1+sqrt(1+t^(2))+c`
`I=int(xdx)/(sqrt(1+x^(2)+sqrt((1+x^(2))^(3))))`
`=int(xdx)/(sqrt((1+x^(2))(1+sqrt(1+x^(2)))))`
` "Let " 1+sqrt(1+x^(2))=t^(2)`
` :. 2tdt =(x)/(sqrt(1+x^(2)))dx`
` :. I=int2t=t+c=1+sqrt(1+t^(2))+c`
Discussion
377.

Evaluate:`int((x-x^3)^(1/3))/(x^4)dx`

Answer» Correct Answer - ` -(3)/(8)((1)/(x^(2))-1)^(4//3)+C`
`I=int((x-x^(3))^(1//3))/(x^(4))dx=int(((1)/(x^(2))-1)^(1//3))/(x^(3))dx`
` "Putting " (1)/(x^(2))=t, (1)/(x^(3))dx= -(dt)/(2), " we get" `
`I= -(1)/(2)int t^(1//3)dt= -(3)/(8)t^(4//3)+C= -(3)/(8)((1)/(x^(2))-1)^(4//3)+C`
Discussion
378.

Evaluate:`int(dx)/(x^2(1+x^5)^(4/5))`

Answer» Correct Answer - ` -((1+x^(5))^(1//5))/(x)+C`
`I=int(dx)/(x^(2)(1+x^(5))^(4//5))=int(dx)/(x^(6)((1)/(x^(5))+1)^(4//5))`
` "Let " t=1+(1)/(x^(5)) " or " dt= -(5dx)/(x^(6))`
` :. I= -(1)/(5) int(dt)/(t^(4//5))= -t^(1//5)+C= -(1+(1)/(x^(5)))^(1//5)+C`
`= -((1+x^(5))^(1//5))/(x)+C`
Discussion
379.

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

Answer» ` I=int(dx)/((1-x^(2))sqrt(1-x^(2)))`
Put `x="sin"theta`
`:. I=int("cos"theta d theta)/((1+sin^(2)theta)(cos theta))`
`=int(d theta)/(1+sin^(2)theta)`
`=int("cosec"^(2)theta d theta)/(2+cot^(2)theta)`
Put `cot theta=t`
`:. -"cosec"^(2)theta d theta=dt`
`:. I=int(-dt)/(2+t^(2))`
`=-(1)/(sqrt(2))"tan"^(-1)(t)/(sqrt(2))+c`
`= -(1)/(sqrt(2))"tan"^(-1)(cot theta)/(sqrt(2))+c`
`=-(1)/(sqrt(2))"tan"^(-1)(sqrt(1-x^(2)))/(sqrt(2x))+c`
Discussion
380.

The value of ` int(1+logx)/(sqrt((x^(x))^(2)-1))dx " is " `A. `sec^(-1)(x^(x))+c`B. `tan^(-1)(x^(x))+c`C. `log(x^(x)+sqrt(x^(2x)-1))+c`D. `cot^(-1)(x^(x))+c`

Answer» Correct Answer - A
`I=int(x^(x)(1+logx))/(x^(x)sqrt((x^(x))^(2)-1))dx `
` "Let " x^(x)=t`
` :. xlogx=log t `
` implies (logx+1)dx=(1)/(t)dt `
` implies x^(x)(1+logx)dx=dt `
` :. I=int(1)/(tsqrt(t^(2)-1))dt `
`=sec^(-1)(t)=sec^(-1)(x^(x))+C`
Discussion
381.

Evaluate:`int((log)_(e x)edot(log)_(e^2)edot(log)_(e^3x)e)/x dx`

Answer» `I=int(log_(ex)e*log_(e^(2)x)e*log_(e^(3)x)e)/(x)dx`
`=int(1)/(xlog_(e)ex*log_(e)e^(2)x*log_(e)e^(3)x)dx`
` =int(1)/(x(log_(e)e+log_(e)x)(log_(e)e^(2)+log_(e)x)(log_(e)e^(3)+log_(e)x))dx`
`=int(1)/((1+t)(2+t)(3+t))dt, " where " t=log_(e)x`
`=int((1)/(2)*(1)/(1+t)-(1)/(2+t)+(1)/(3+t))dt " " `[Using partial fractions]
`=(1)/(2)log|1+t|-log|2+t|+log|3+t|+C`
`=(1)/(2)log|1+log_(e)x|-log|2+log_(e)x|+log|3+log|3+log_(e)x|+C`
Discussion
382.

Evaluate:`int(2x+1)/(x^4+2x^3+x^2-1)dx`

Answer» `I = int(2x+1)/(x^(4)+2x^(3)+x^(2)-1)dx`
`=int(2x+1)/((x^(2)+x)^(2)-1)dx`
Let `x^(2)+x=t`
`:. dt=(2x+1)dx`
`:. I=int(dt)/(t^(2)-1)`
`=(1)/(2)log_(e)|(x^(2)+x-1)/(x^(2)+x+1)|+C`
Discussion
383.

Evaluate:`int(dx)/(x^2(x^4+1)^(3/4))`

Answer» `I=int(dx)/(x^(2)(x^(4)+1)^((3)/(4)))=int(dx)/(x^(2).x^(3)(1+(1)/(x^(4)))^((3)/(4)))`
`=int((1+(1)/(x^(4)))^(-(3)/(4))dx)/(x^(5))`
Let `1+(1)/(x^(4))=t " or " (-4)/(x^(5))dx=dt`
` :. I= -(1)/(4)int t^(-(3)/(4))dt= -(1)/(4)(t^((1)/(4)))/((1)/(4))+C`
`= -(1+(1)/(x^(4)))^((1)/(4))+C`
Discussion
384.

`int(x^(2)(1-logx))/((logx)^(4)-x^(4))dx` equalsA. `(1)/(2)ln.(x)/(lnx)-(1)/(4)ln(ln^(2)x-x^(2))+C`B. `(1)/(4)ln((lnx-x)/(lnx+x))-(1)/(2)tan^(-1)((lnx)/(x))+C`C. `(1)/(4)ln((lnx-x)/(lnx+x))+(1)/(2)tan^(-1)((lnx)/(x))+C`D. `(1)/(4)(ln((lnx-x)/(lnx+x))+tan^(-1)((lnx)/(x)))+C`

Answer» Correct Answer - B
`I=int(x^(2)(1-logx))/((logx)^(4)-x^(4))dx`
`=int(1-logx)/(x(((logx)/(x))^(4)-1))dx`
Put `(logx)/(x)=t rArr (1-logx)/(x^(2))=dt`
`therefore" "I=int(dt)/((t^(4)-1))=int(dt)/((t^(2)+1)(t^(2)-1))`
`=(1)/(2)int((t^(2)+1)-(t^(2)-1))/((t^(2)+1)(t^(2)-1))dt`
`=(1)/(2)(int(dt)/(t^(2)-1)-int(dt)/(t^(2)+1))=(1)/(2)((1)/(2)ln(t-1)/(t+1)-tan^(-1)t)`
`=(1)/(4)ln((lnx-x)/(lnx+x))-(1)/(2)tan^(-1)((lnx)/(x))+C`
Discussion
385.

Evaluate:`int(dx)/((x-p)sqrt((x-p)(x-q)))`

Answer» Correct Answer - `-(2)/(p-q)sqrt((x-q)/(x-p))+C`
`sqrt((x-q)/(x-p))=t`
` :. (1)/(2)((x-p)/(x-q))^(1//2)((x-p)1-(x-q)1)/((x-p)^(2))dx=dt.`
`or (1)/(2)(q-p)/(sqrt(x-q)(x-p)^(3//2))dx=dt`
`or (dx)/((x-p)^(3//2)sqrt((x-q)))=(2dt)/(q-p)`
`or I=-int(2dt)/(p-q)t= -(2)/(p-q)sqrt((x-q)/(x-p))+C`
Discussion
386.

Evaluate: `int((x^4-x)^(1//4))/(x^5) dx`

Answer» Correct Answer - `(4)/(15)(1-(1)/(x^(3)) )^(5//4)+C`
`int((x^(4)-x)^(1//4))/(x^(5))dx=int(1)/(x^(4))(1-(1)/(x^(3)))^(1//4)dx,`
` "Putting "1-(1)/(x^(3))=t, " we get" `
`I=(1)/(3)int t^(1//4)dt =(4)/(15)t^(5//4)+C=(4)/(15)(1-(1)/(x^(3)) )^(5//4)+C`
Discussion
387.

Evaluate:`intx/((x-1)(x^2+4))dx`

Answer» `(x)/((x-1)(x^(2)+4))dx`
Let `(x)/((x-1)(x^(2)+4))=(A)/(x-1)+(Bx+C)/(x^(2)+4) " " ` (1)
or `x=A(x^(2)+4)+(Bx+C)(x-1) " " `(2)
Putting `x=1` in equation (2), we get `I=5A`.
Putting `x=0` in equation (2), we get `0=4A-C.`
Putting `x= -1` in equation (2), we get `-1=5A+2B-2C`.
Solving these equations, we obtain ` A=(1)/(5), B= -(1)/(5), " and " C=(4)/(5)`.
Substituting the values of A, B, and C in equation (1), we obtain
`(x)/((x-1)(x^(2)+4))=(1)/(5(x-1))+(-(1)/(5)x+(4)/(5))/(x^(2)+4)`
`=(1)/(5(x-1))-(1)/(5)((x-4))/((x^(2)+4))`
or `I=(1)/(5)int(1)/(x-1)dx-(1)/(5)int(x-4)/(x^(2)+4)dx`
`=(1)/(5)int(1)/(x-1)dx-(1)/(10)int(2x)/(x^(2)+4)dx+(4)/(5)int(1)/(x^(2)+4)dx`
`=(1)/(5)log|x-1|-(1)/(10)log(x^(2)+4)+(4)/(5)xx(1)/(2)"tan"^(-1)(x)/(2)+C`
`=(1)/(5)log|x-1|-(1)/(10)log(x^(2)+4)+(2)/(5)"tan"^(-1)((x)/(2))+C`
Discussion
388.

Evaluate:`intx/(x^4+2x^2+3)dx`

Answer» ` I=int(x)/(x^(4)+2x^(2)+3)dx`
Let `x^(2)=t.`
` :. 2x dx=dt`
` :. I=(1)/(2)int(dt)/(t^(2)+2t+3)`
`=(1)/(2)int(dt)/((t+1)^(2)+2)`
`=(1)/(2)(1)/(sqrt(2))"tan"^(-1)(t+1)/(sqrt(2))+c`
`=(1)/(2sqrt(2))"tan"^(-1)(x^(2)+1)/(sqrt(2))+c`
Discussion
389.

Evaluate `int(dx)/(x^(2)sqrt(1+x^(2)))`.

Answer» `I=int(dx)/(x^(2)sqrt(1+x^(2)))=int(dx)/(x^(3)sqrt(1+(1)/(x^(2))))`
Let `t=sqrt(1+(1)/(x^(2))) " or " (dt)/(dx)=(1(-(2)/(x^(3))))/(2sqrt(1+(1)/(x^(2))))`
or `(dx)/(x^(3))= -t dt`
or `I= -int(tdt)/(t)= -t +C= -sqrt(1+(1)/(x^(2))) +C`
`= - (1)/(x)sqrt(1+x^(2))+C`
Discussion
390.

Evaluate:Evaluate:`intx^(-11)(1+x^4)^(-1/2)dx`

Answer» `I=int (dx)/(x^(11)(1+x^(4))^(1//2))=int (dx)/(x^(11)*x^(2)(1+1//x^(4))^(1//2))`
Let `1+(1)/(x^(4))=t^(2) " or " (-4)/(x^(5))dx=2tdt`
`:. I=int(dx)/(x^(13)(1+1//x^(4))^(1//2 ))`
`=-(1)/(4)int(2tdt)/(x^(8)t)`
`=-(1)/(2)int(t^(2)-1)^(2)dt`
`=-(1)/(2)int(t^(4)-2t^(2)+1)dt`
`= -(1)/(2)[(t^(5))/(5)-(2t^(3))/(3)+t]+C, " where " t=sqrt(1+(1)/(x^(4)))`
Discussion
391.

If `I=intx^(27)(6x^(2)+5x+4)(x^(2)+x+1)^(6)dx=f(x)+C`, then f(x) is equal toA. `(x^(4)(1+x+x^(2))^(7))/(7)+C`B. `(x^(28)(1+x+x^(2))^(7))/(7)+C`C. `(x^(28)(1+x+x^(2))^(7))/(28)+C`D. None

Answer» Correct Answer - B
`I=intx^(24)(1+x+x^(2))^(6).{x^(3)(4+5x+6x^(2))}dx`
`=int(x^(4)+x^(5)+x^(6))^(6).(6x^(5)+5x^(4)+4x^(3))dx`
`=(x^(28)(1+x+x^(2))^(7))/(7)+c`
Discussion
392.

Evaluate:`intx^x1n(e x)dx`

Answer» Correct Answer - `x^(x)+C`
`I=int x^(x) " In "(ex)dx=int x^(x) (1+"In "x)dx`
` "Let "t=x^(x)=e^(x"In"x)" then " (dt)/(dx)=x^(x)(x xx (1)/(x)+"In "x)`
` or dt=x^(x)(1+"In "x)dx or I=intdt=t+C=x^(x)+C`
Discussion
393.

If `l=intsec^2xcos e c^4x dx=Acot^3x+Btanx+Ccotx+D ,`then`A=-1/3`(b) `B=2``C=-2`(d) none of theseA. `A=-(1)/(3)`B. `B=2`C. `C=-2`D. none of these

Answer» Correct Answer - A::C
`I=int sec^(2)x "cosec"^(4)xdx`
`=int((sin^(2)x+cos^(2)x)^(2))/(cos^(2)xsin^(4)x)dx`
`=int(sin^(4)x+cos^(4)x+2sin^(2)x cos^(2)x)/(cos^(2)xsin^(4)x)dx`
`=int(sec^(2)x+2"cosec"^(2)x+(cos^(2)x)/(sin^(4)x))dx `
`=tanx-2cotx+int cot^(2)x"cosec"^(2)x dx`
`=tanx-2cotx-(cot^(3)x)/(3)+D`
Discussion
394.

If `int 1/(1-sin^4x)dx=1/(asqrtb)+tan^(- 1)(sqrt(a)tanx)+1/btanx+C` then `a/b` is equal to

Answer» Correct Answer - 1
Discussion
395.

What is `int e^(x ln (a)) dx` equal to ?A. `(a^(x))/(ln (a)) + c`B. `(e^(x))/(ln (a)) + c`C. `(e^(x))/(ln (ae)) + c`D. `(ae^(x))/(ln (a)) + c`

Answer» Correct Answer - A
`int e^(x ln (a)).dx = int e^(x) .dx = (a^(x))/(ln (a)) + c`
Discussion
396.

Evaluate:`intsin^2(logx)dx`

Answer» Correct Answer - `(1)/(10)x(5-2sin(2logx)-cos(2logx))+C`
`I=int sin^(2)(logx)dx`
` "Let "t=log x " or "dt=(dx)/(x)`
`"or " dx=e^(t)dt`
`:. I=int e^(t) sin^(2)t dt=(1)/(2)int e^(t)(1-cos2t)dt`
`"or "2I=e^(t)-int e^(t)cos2t dt`
`= e^(t)-(e^(t))/(5)(2sin2t+cos2t)+C`
`"or " I=(1)/(10)x(5-2sin(2logx)-cos(2logx))+C`
Discussion
397.

For any natural number m, evaulate, `int(x^(3m)+x^(2m)+x^(m))(2x^(2m)+3x^m+6)^(1//m)dx, x gt0`A. `(x^(2)-1)/(x^(3)sqrt(2x^(4)-2x^(2)+1)+C`B. `sqrt(2x^(4)-2x^(2)+1)/(x^(3))`C. `sqrt(2x^(4)-2x^(2)+1)/(x)+C`D. `sqrt(2x^(4)-2x^(2)+1)/(2x^(2))+C`

Answer» `(2x^(3m)+3x^(2m)+6x^(m))^((m+1)/(m))/(6(m+1))+C`
Discussion
398.

Evaluate:`intsin^(-1)((2x+2)/(sqrt(4x^2+8x+13)))dx`

Answer» `(x+1)tan(2x+2)/(3)-3/4"ln"(4x^(2)+8x+13)+C`
Discussion
399.

Evaluate:`int((1+x)^3)/(sqrt(x))dxdot`

Answer» `int((1+x)^(3))/(sqrt(x))dx=int(1+3x+3x^(3)+x^(3))/(sqrt(x))dx`
`=intx^(-1//2)dx +3intx^(1//2)dx+3intx^(3//2)dx+intx^(5//2)dx`
`=(x^(1//2))/(1//2)+3(x^(3//2))/(3//2)+3*(x^(5//2))/(5//2)+(x^(7//2))/(7//2)+c`
`=2sqrt(x)+2x^(3//2)+(6)/(5)x^(5//2)+(2)/(7)x^(7//2)+c`
Discussion
400.

Let`f(x)=inte^x(x-1)(x-2)dxdot`Then `f`decreases in the interval`(-oo,-2)`(b) `-2,-1)``(1,2)`(d) `(2,+oo)`A. `-(infty,2)`B. `(-2,-1)`C. (1,2)D. `(2,+infty)`

Answer» Correct Answer - C
Discussion