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151.

If `int ((x^2-x+1)/(x^2+1) ) e^(cot^(-1)x dx)=A(x) e^(cot^(-1)x)+c, A=`A. `-x`B. xC. `sqrt(1-x)`D. `sqrt(1+x)`

Answer» Correct Answer - B
`LHS=int[(x^(2)+1)/(x^(2)+1)-(x)/(x^(2)+1)]e^(cot^(-1)x)dx`
`=int1.e^(cot^(-1)x)dx-int(1)/(x^(2)+1)e^(cot^(-1)x)dx`
On integration by parts, we get
`xe^(cot(-1)x)-int x.e^(cot^(-1)x)(-(1)/(1+x^(2)))dx-int(x)/(1+x^(2))e^(cot^(-1)x)dx+C`
`=xe^(cot^(-1)x)+C`
Discussion
152.

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

Answer» Correct Answer - `(1)/(-6(x^(3) +4)^(2))+c`
Discussion
153.

If `l_(n)=intx^(n).e^(cx)dx` for `n ge 1`, then `C.l_(n)+n.l_(n-1)` is equal toA. `x^(n)e^(cx)`B. `x^(n)`C. `e^(cx)`D. `x^(n)+e^(cx)`

Answer» Correct Answer - A
Given, `l_(n)=int x^(n).e^(cx)dx=x^(n).(e^(cx))/(c)-int nx^(n-1).(e^(cx))/(c)dx`
`rArr" "l_(n)=(e^(cx).x^(n))/(c)-(n)/(c)l_(n-1)`
`rArr" "cl_(n)+nl_(n-1)=e^(cx).x^(n)`
Discussion
154.

`int32x^(3)(logx)^(2)` dx is equal otA. `8x^(4)(logx)^(2)+C`B. `x^(4){8(logx)^(2)-4log x+1}+C`C. `x^(4){8(logx)^(2)-4logx}+C`D. `x^(3){(logx)^(2)+2logx}+C`

Answer» Correct Answer - B
`int 32x^(3)(logx)^(2)dx`
On using integration by parts, we get
`32{(logx)^(2).(x^(4))/(4)-int 2 log x.(1)/(x).(x^(4))/(4)dx}`
`=8x^(4)(logx)^(2)-16intx^(3)log x dx`
Again using integration by parts, we get
`8x^(4)(logx)^(2)-16{logx.(x^(4))/(4)-int(1)/(x).(x^(4))/(4)dx}`
`=8x^(4)(logx)^(2)-4x^(4)logx+4int x^(3)dx`
`=8x^(4)(logx)^(2)-4x^(4)logx+x^(4)+C`
`=x^(4)[8(logx)^(2)-4logx+1]+C`
Discussion
155.

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

Answer» Correct Answer - `(1)/(sqrt(5))log |(sqrt(5)+2x+1)/(sqrt(5)-2x-1)|+c`
Discussion
156.

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

Answer» Correct Answer - `(1)/(2sqrt(2)) log |(sqrt(2x)-sqrt(2)-1)/(sqrt(2x)-sqrt(2)+1)|+c`
Discussion
157.

The value of the integral `intx sin^(-1)xdx` is equal toA. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`B. `(1)/(2)x^(2)sin^(-1)x-(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`C. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))+(1)/(4)sin^(-1)x+C`D. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)sqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`

Answer» Correct Answer - A
Let `l=int x sin^(-1)xdx`
On using integration by parts, we get
`l=(sin^(-1)x)(x^(2))/(2)-int(1)/(sqrt(1-x^(2))).(x^(2))/(2)dx`
`rArr" "l=(x^(2))/(2)sin^(-1)x+(1)/(2)int(-x^(2))/(sqrt(1-x^(2)))dx`
`=(x^(2))/(2)sin^(-1)x+(1)/(2)int(1-x^(2)-1)/(sqrt(1-x^(2)))dx`
`rArr" "=(x^(2))/(2)sin^(-1)x+(1)/(2){int(1-x^(2))/(sqrt(1-x^(2)))dx-int(1)/(sqrt(1-x^(2)))dx}`
`rArr ,=(x^(2))/(2)sin^(-1)x+(1)/(2){int sqrt(1-x^(2))dx-int(1)/(sqrt(1-x^(2)))dx}`
`=(x^(2))/(2)sin^(-1)x+(1)/(2)[{(1)/(2)xsqrt(1-x^(2))+(1)/(2)sin^(-1)x}-sin^(-1)x]+C`
`rArr l=(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`
Discussion
158.

`int(4x-3)/(3x^2+2x-5)dx`

Answer» Correct Answer - `(2)/(3) log |3x^(2)+2x-5|-(13)/(24)log |(3x-3)/(3x+5)|+c`
Discussion
159.

`int[sin(logx)+cos(logx)]dx`A. `x log (logx)+C`B. `sin(logx)+C`C. `cos(logx)+C`D. `x sin (logx)+C`

Answer» Correct Answer - D
`int[sin(logx)+cos(logx)]dx`
`=intsin(logx)dx+intcos(logx)dx`
`=x sin (logx)-int(xcos (logx))/(x)dx+intcos(logx)dx+C`
`=x sin (logx)+C`
Discussion
160.

`int x/(x+a)dx`

Answer» Correct Answer - `x-a log (x+a) +c`
Discussion
161.

If `l_(1)=int sin^(-1)x dx` and `l_(2) =int sin^(-1)sqrt(1-x^(2))dx`, thenA. `l_(1)=l_(2)`B. `l_(2)=(pi)/(2)l_(1)`C. `l_(1)+l_(2)=(pi)/(2)x`D. `l_(1)+l_(2)=(pi)/(2)`

Answer» Correct Answer - C
Given, `l_(1)=int sin^(-1)xdx`
and `l_(2)=int sin^(-1)sqrt(1-x^(2))dx rArr l_(2)=int cos^(-1)xdx`
Now, `l_(1)+l_(2)=int(sin^(-1)x+cos^(-1)x)dx=int(pi)/(2)dx=(pi)/(2)x`
`therefore" "l_(1)+l_(2)=(pi)/(2)x`
Discussion
162.

Evaluate:`int1/(sinx(3+2cosx)dx`

Answer» Correct Answer - `(1)/(10) log |1-cos x|-(1)/(2)log |1+ cos x|+(2)/(5) log | 3+2 cos x|+c`
Discussion
163.

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

Answer» Correct Answer - `(2)/(3) x^(5//2) +(4)/(3)x^(3//2) -10 sqrt(x)+c`
Discussion
164.

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)/(sqrt3)tan^(-1)((2sqrt(x-1)+1)/(sqrt3))+C`D. None of the above

Answer» Correct Answer - C
Let `l=int(dx)/(x+sqrt(x-1))`
Put`" "x=t^(2)+1 rArr dx=2tdt`
`therefore" "l=int(2t)/(t^(2)+t+1)dt=int(2t+1)/(t^(2)+t+1)dt-int(1)/(t^(2)+t+1)dt`
`=log(t^(2)+t+1)-int(1)/((t+(1)/(2))^(2)+((sqrt3)/(2))^(2))dt`
`=log(t^(2)+t+1)-(2)/(sqrt3)tan^(-1)((2t+1)/(sqrt3))`
`=log(x+sqrt(x-1))-(2)/(sqrt3)tan^(-1)((2sqrt(x-1)+1)/(sqrt3))+C`
Discussion
165.

`intsqrt((x-1)/(x+1))dx` is equal toA. `2sqrt(x^(2)+1)+sin^(-1)x+C`B. `sqrt(x^(2)-1)-sin^(-1)x+C`C. `sqrt(x^(2)-1)+sin^(-1)x+C`D. `(sqrt(x^(2)-1))/(2)+sin^(-1)x+C`

Answer» Correct Answer - B
Let `l=intsqrt((x-1)/(x+1))dx`
`rArr" "l=int(x-1)/(sqrt(x^(2)-1))dx=int(x)/(sqrt(x^(2)-1))dx-int(1)/(sqrt(x^(2)-1))dx`
`=sqrt(x^(2)-1)-sin^(-1)x+C`
Discussion
166.

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

Answer» Correct Answer - D
Let `l=int(e^(x)+e^(-x))^(2).(e^(x)-e^(-x))dx`
Put `e^(x)+e^(-x)=t rArr (e^(x)-e^(-x))dx=dt`
`therefore" "l=int t^(2)dt=(t^(3))/(3)+C=((e^(x)+e^(-x))^(3))/(3)+C`
Discussion
167.

Evaluate: `int1/(x (x^4+1)) dx`

Answer» Correct Answer - `(1)/(2) log |(x^(4))/(x^(4)+1)|+c`
Discussion
168.

`int(tan x + cos x)^(2) dx `

Answer» Correct Answer - `tan x=cot x+c`
Discussion
169.

`intcos^(3)xe^(log(sinx))dx` is equal toA. `(-sinn^(4)x)/(4)+C`B. `-(cos^(4)x)/(4)+C`C. `(e^(sinx))/(4)+C`D. None of these

Answer» Correct Answer - B
Let `l=int cos^(3)x e^(log sinx)dx=int cos^(3)x sin x dx`
Put `cos x = t rArr - sin x dx = dt`
`therefore" "l=-int t^(3)dt=-(t^(4))/(4)+C=-(cos^(4)x)/(4)+C`
Discussion
170.

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

Answer» Correct Answer - `(1)/(2) log | 1+x|-(1)/(4) log |1+x^(2)|+(1)/(2)tan^(-1) x+c`
Discussion
171.

`int(sinx-cosx)^(4)(sinx+cosx)dx` is equal toA. `(sin x - cosx)/(5)+C`B. `((sin x - cosx)^(5))/(5)+C`C. `((sin x-cosx)^(4))/(4)+C`D. `((sinx+cosx)^(5))/(5)+C`

Answer» Correct Answer - B
Let `l=int(sinx-cosx)^(4)(sinx +cosx)dx`
Put`" "Sinx-cosx=t`
`rArr" "(cos x+sin x)dx=dt`
`therefore" "l=int t^(4)dt=(t^(5))/(5)+C=((sinx-cosx)^(5))/(5)+C`
Discussion
172.

` sqrt(sin 2x) cos 2x`A. `(1)/(3)(sinx)^(3//2)+C`B. `(1)/(3)(sin x)^(1//2)+C`C. `(1)/(3)(sin 2x)^(3//2)+C`D. `(1)/(3)(sin 2x)^(1//2)+C`

Answer» Correct Answer - C
`intsqrt(sin2x)cos 2xdx`
Let `" "sin2x=t" "rArr" "2cos 2x=(dt)/(dx)`
`rArr" "dx=(dt)/(2cos 2x)`
`therefore int sqrt(sin2x)cos 2x dx=intsqrtt cos 2x(dt)/(2cos 2x)=(1)/(2)int sqrtt dt`
`=(1)/(2)(t^(1//2+1))/(((1)/(2)+1))+C=(1)/(3)t^(3//2)+C`
`=(1)/(3)(sin2x)^(3//2)+C`
Discussion
173.

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

Answer» Correct Answer - `(1)/(2) tan x+c`
Discussion
174.

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

Answer» Correct Answer - D
Let `l=int(1+tan^(2)x)/(1-tan^(2)x)dx=int(sec^(2)x)/(1-tan^(2)x)dx`
Put`" "tan x= t rArr sec^(2)x dx=dt`
` therefore" "l=int(dt)/(1-t^(2))=(1)/(2xx1)log((1+t)/(1-t))+C`
`=(1)/(2)log((1+tanx)/(1-tanx))+C`
Discussion
175.

`(i) int (1)/(x(x+1)^(2))dx` `(ii) int (1)/((x+1)^(2)(x-1))dx`

Answer» Correct Answer - `(i) (1)/(x+1) -log |(x+1)/(x)|+c`
`(ii) (1)/(4)log |(x-1)/(x+1)|+(1)/(2(x+1))+c`
Discussion
176.

If `l=int(x^(5))/(sqrt(1+x^(3)))dx`, then l is equal toA. `(2)/(9)(1+x^(3))^((5)/(2))+(2)/(3)(1+x^(3))^((3)/(2))+C`B. `log|sqrtx+sqrt(1+x^(3))|+C`C. `log|sqrtx-sqrt(1+x^(3))|+C`D. `(2)/(9)(1+x^(3))^((3)/(2))-(2)/(3)(1+x^(3))^((1)/(2))+C`

Answer» Correct Answer - D
Given, `" "l=int(x^(5))/(sqrt(1+x^(3)))dx`
Let`" "1+x^(3)=t`
`rArr" "3x^(2)dx=dt`
`therefore" "l=int((t-1))/(sqrtt).(dt)/(3)=(1)/(3)int(sqrtt-t^(-1//2))dt`
`=(1)/(3)[(2t^(3//2))/(3)-2t^(1//2)]+c`
`=(2)/(9)(1+x^(3))^(3//2)-(2)/(3)(1+x^(3))^(1//2)+C`
Discussion
177.

`intsqrt(1+cos 2x) dx`

Answer» Correct Answer - `sqrt(2) sin x+c`
Discussion
178.

`int(1)/(1-cos 2x) dx`

Answer» Correct Answer - `-(1)/(2) cot x+c`
Discussion
179.

`int_(a)^(2a) (sqrt((a)/(x))+sqrt((x)/(a)))^(2)dx`

Answer» Correct Answer - `a log 2 + (7a)/(2)`
Discussion
180.

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

Answer» Correct Answer - `log |x+1|+(1)/(x+1)+c`
Discussion
181.

`intcos{2tan^(-1)sqrt((1-x)/(1+x))}dx` is equal toA. `(1)/(8)(x^(2)-1)+C`B. `(x^(2))/(4)+C`C. `(x)/(2)+C`D. `(x^(2))/(2)+C`

Answer» Correct Answer - D
Let `l=intcos{2tan^(-1)sqrt((1-x)/(1+x))}dx`
Put`" "x= cos theta,` then
`l=int cos {2tan^(-1)sqrt((1-cos theta)/(1+cos theta))}dx`
`=int cos{2tan^(-1)(tan.(theta)/(2))}dx`
`=intcos theta dx=int xdx=(x^(2))/(2)+C`
Discussion
182.

`int(x^(2))/(1+x^(6)) dx`

Answer» Correct Answer - `(1)/(3) tan^(-1) x^(3) +c`
Discussion
183.

`int(x^2)/((a+b x)^2)dx`

Answer» Correct Answer - `(1)/(b^(3)) [(a +bx) -2a log (a +bx)-(a^(2))/(a+bx)]+c`
Discussion
184.

`int((log _(e)x)^(3))/(x)dx`

Answer» Correct Answer - `(1)/(4) (log_(e)x)^(4) +c`
Discussion
185.

`int dx/(sin^2 x cos^2 x)` equalsA. `tan x-cot x+c`B. `tan x cot x+c`C. `tan x- cot 2x+c`D.

Answer» Correct Answer - b
` int (dx)/(sin^(2) x cos^(2)x)= int (sin^(2) x+cos ^(2) x)/(sin^(2) x cos^(2) x)dx`
`= int (sin^(2)x)/(sin^(2) x cos^(2)x) dx+ int (cos^(2)x)/(sin^(2) x cos ^(2) x)dx`
` = int sec^(2) x dx + int " cosec"^(2) x dc`
` = tan x - cot x+c`
Discussion
186.

`int_(1)^(e) (e^(x) (1+xlog x))/(x) dx`

Answer» Correct Answer - `e^(e)`
Discussion
187.

`int_(0)^(pi//2) (1)/(4+3 cos x)dx`

Answer» Correct Answer - `(2)/(sqrt(7)) tan^(-1) .(1)/(sqrt(7))`
Discussion
188.

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

Answer» Correct Answer - `pi+2`
Discussion
189.

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

Answer» Correct Answer - `(1)/(2) (3-sqrt(3))`
Discussion
190.

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

Answer» Correct Answer - `(sqrt(e)-1)/(e)`
Discussion
191.

`int_(1)^(2) (1)/(2(1+x^(4)))dx`

Answer» Correct Answer - `(1)/(4) log (32)/(17)`
Discussion
192.

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

Answer» Correct Answer - `(2)/(sqrt(7))[tan^(-1) (5)/(sqrt(7)) -tan^(-1) .(1)/(sqrt(7))]`
Discussion
193.

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

Answer» Correct Answer - 1
Discussion
194.

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

Answer» Correct Answer - `(3)/(20)`
Discussion
195.

`int_(pi//6)^(pi//2) cos x dx`

Answer» Correct Answer - `(1)/(2)`
Discussion
196.

`int_(0)^(2) (1)/(4+x+x^(2))dx`

Answer» Correct Answer - `(1)/(sqrt(17))log |(21+5sqrt(17))/(4)|`
Discussion
197.

`int_(0)^(pi) sin 3x dx`

Answer» Correct Answer - `(2)/(3)`
Discussion
198.

`int_(0)^(pi//4) sin ^(2) x dx`

Answer» Correct Answer - `(pi -2)/(3sqrt(3))`
Discussion
199.

`(i) int_(0)^(pi//4) e^(tanx) . sec^(2) x dx` `(ii) int_(0)^(pi//4) (sin (cos 2x))/(" cosec " 2x)dx`

Answer» Correct Answer - `(i) (e-1) (ii) (1)/(2) (1- cot 1)`
Discussion
200.

`"F i n d"int_4^9sqrt(x)dx`

Answer» Correct Answer - `(38)/(3)`
Discussion