InterviewSolution
Saved Bookmarks
InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 201. |
`int_(0)^(1//sqrt(2)) (sin^(-1))/((1-x^(2))^(3//2))dx` |
| Answer» Correct Answer - `(pi)/(4)-(1)/(2)log 2` | |
| 202. |
`int_(0)^(a) y^(2) dx` |
| Answer» Correct Answer - `(2)/(3) a^(3)` | |
| 203. |
`int_(1)^(2) (1)/(x(1+x))dx` |
| Answer» Correct Answer - `log (4)/(3)` | |
| 204. |
`int_(0)^(pi//2) x^(2) cos x dx` |
| Answer» Correct Answer - `((pi^(2))/(4)-2)` | |
| 205. |
`int_(0)^(pi//6) sqrt(1-sin 2x) dx` |
| Answer» Correct Answer - `(1)/(2) (sqrt(3)-1)` | |
| 206. |
`int_(0)^(pi//2) sqrt(1- cos 2x) dx` |
| Answer» Correct Answer - `(pi)/(3)` | |
| 207. |
`int_(0)^(1) (1)/(sqrt(1-x^(2)))dx` |
| Answer» Correct Answer - `(pi)/(2)` | |
| 208. |
`int_(0)^(pi//2) cos 3x dx` |
| Answer» Correct Answer - `-(1)/(3)` | |
| 209. |
`int_1^2 (x+3)/(x(x+2))dx` |
| Answer» Correct Answer - `(1)/(2) log 6` | |
| 210. |
`int_(0)^(pi//2) sin ^(4) x dx` |
| Answer» Correct Answer - `(3pi)/(16)` | |
| 211. |
`int_(0)^(1) x (1 -x)^(n) dx=?`A. `(1)/((n+1)(n+2))`B. `(1)/(n(n+2))`C. `(1)/((n+1)(n+3))`D. |
| Answer» Correct Answer - B | |
| 212. |
`int (x^(2))/(x^(2) +4) dx=?`A. `x+4 tan^(-1) ((x)/(2))+c`B. `x-2 tan^(-1) ((x)/(2))+c`C. None of the aboveD. |
| Answer» Correct Answer - C | |
| 213. |
`int_(-pi//4)^(pi//4) "cosec"^(2) x dx` |
| Answer» Correct Answer - `-2` | |
| 214. |
`int_(0)^(pi//2) (dx)/(4sin^(2) x + 5 cos^(2)x)` |
| Answer» Correct Answer - `(pi)/(4sqrt(5))` | |
| 215. |
`int_(0)^(1) (1)/(x^(2)+x+1)dx` |
| Answer» Correct Answer - `(pi)/(3sqrt(3))` | |
| 216. |
`int_(0)^(1) (1)/(e^(x) +e^(-x)) dx` |
| Answer» Correct Answer - `tan^(-1) e-(pi)/(4)` | |
| 217. |
`int_(-2)^(2) X sin^(10) x dx =?`A. `-2`B. 4C. None of the aboveD. |
| Answer» Correct Answer - D | |
| 218. |
`int_(2)^(8) |x-5| dx=?`A. 9B. 10C. 11D. |
| Answer» Correct Answer - B | |
| 219. |
`int_(pi//6)^(pi//2) (" cosec x cot x")/(1+" cosec "^(2) x)dx` |
| Answer» Correct Answer - `tan^(-1) .(1)/(3)` | |
| 220. |
`(i) int_(0)^(pi//2) x sin x cos x dx` `(ii) int_(0)^(pi//6) (2+3x^(2)) cos 3x dx` |
| Answer» Correct Answer - `(i) (1)/(36) (pi^(2) +16) (ii) (pi)/(8)` | |
| 221. |
`int_(0)^(pi) (1)/(5+2 cos x)dx` |
| Answer» Correct Answer - `(pi)/(sqrt(21))` | |
| 222. |
`int_(-4)^(4) log ((7-x )/(2+x)) dx=?`A. 4B. -4C. 0D. 1 |
| Answer» Correct Answer - C | |
| 223. |
`int_(1)^(2) (dx)/(x(1+log x)^(2))` |
| Answer» Correct Answer - `(log 2)/(1+log 2)` | |
| 224. |
`int_(0)^(pi//4) tan x dx` |
| Answer» Correct Answer - `(1)/(2) log 2` | |
| 225. |
`(i) int_(0)^(pi//2) x cos x dx` (i) `int_(1)^(3) x. log x dx` |
| Answer» Correct Answer - `(i) (pi)/(2)-1 (ii) (9)/(2) log 3-2` | |
| 226. |
`int_(0)^(pi//4) e^(x) (tan x+ sec^(2)x) dx` |
| Answer» Correct Answer - `e^(pi//4)` | |
| 227. |
`int_(0)^(pi) (1-x^(2))/((1+x^(2))^(2))dx` |
| Answer» Correct Answer - `(1)/(2)` | |
| 228. |
`int_(0)^(pi//4) log (1+tan x) dx =?`A. `(pi)/(4) log 2`B. `(pi)/(6) log 2`C. `(pi)/(8) log 2`D. |
| Answer» Correct Answer - D | |
| 229. |
`int_(0)^(pi//2) e^(x) (sin x + cos x) dx` |
| Answer» Correct Answer - `e^(pi//2)` | |
| 230. |
`int_(0)^(1) (1)/(x^(2) +2x+3)dx` |
| Answer» Correct Answer - `(1)/(sqrt(2)) tan^(-1) (1)/(2sqrt(2))` | |
| 231. |
`int_(0)^(pi) x sin^(2) x dx` |
| Answer» Correct Answer - `(pi^(2))/(4)` | |
| 232. |
prove that `int_0^oo x^2/((x^2+a^2)(x^2+b^2))dx=pi/(2(a+b))` |
| Answer» Correct Answer - `(pi)/(2(a+b))` | |
| 233. |
`int_(1)^(2) (log_(e) x)/(x^(2)) dx` |
| Answer» Correct Answer - `(1)/(2) (1-log_(e)2)` | |
| 234. |
`int("cosec x")/(cos^(2)(1+log tan.(x)/(2)))dx` is equal toA. `sin^(2)[1+log tan.(x)/(2)]+C`B. `tan[1+log tan.(x)/(2)]+C`C. `sec^(2)[1+log tan.(x)/(2)]+C`D. `-tan[1+log tan.(x)/(2)]+C` |
|
Answer» Correct Answer - B Let `l=int("cosec x")/(cos^(2)(1+log tan.(x)/(2)))dx` Put `1+log tan.(x)/(2)=t` `rArr (1)/(tan.(x)/(2)).sec^(2).(x)/(2).(1)/(2)dx=dt rArr" cosec x dx = dt "` ltBrgt `therefore" "l=int(dt)/(cos^(2)t)=int sec^(2)t dx=tant+C` `=tan(1+log tan.(x)/(2))+C` |
|
| 235. |
The value of `sqrt(2)int(sinx)/(sin(x-(pi)/(4)))dx` , isA. `x-log|cos(x-(pi)/(4))||+C`B. `x+log|cos(x-(pi)/(4))|+C`C. `x-log|sin(x-(pi)/(4))|+C`D. `x+log|sin(x-(pi)/(4))|+C` |
|
Answer» Correct Answer - D Let `l=sqrt2 int(sinx)/(sin(x-(pi)/(4)))dx` Put `x-(pi)/(4)=t rArr dx=dt` `therefore" "l=sqrt2 int(sin((pi)/(4)+t)dt)/(sint)` `=sqrt2 int(sin .(pi)/(4)cost+cos.(pi)/(4)sint)/(sint)dt` `=sqrt2 int((1)/(sqrt2)(cost)/(sint)+(1)/(sqrt2))dt` `=int(cot +1)dt=log|sint|+t+C_(1)` `=x +log|sin(x-(pi)/(4))|+C" "[becauseC_(1)-(pi)/(4)=C]` |
|
| 236. |
`int (x)/( 1+cos x) dx=?`A. `x tan (x)/(2) + 2 log cos (x)/(2) + c`B. ` x tan (x)/(2) + 2 log sin (x)/(2)+c`C. None of the aboveD. |
| Answer» Correct Answer - B | |
| 237. |
`int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2)))dx`A. `log|(sqrt(2xe^(sinx)+1)-1)/(sqrt(2xe^(sinx)+1)+1)|+C`B. `log|(sqrt(2xe^(sinx)-1)+1)/(sqrt(2xe^(sinx)+1)+1)|+C`C. `log|(sqrt(2xe^(sinx)+1)+1)/(sqrt(2xe^(sinx)-1)+1)|+C`D. `log|(sqrt(2xe^(sinx)+1)+1)/(sqrt(2xe^(sinx)-1)-1)|+C` |
|
Answer» Correct Answer - A `l=int((x cos x+1)e^(sinx)dx)/(e^(sinx)x sqrt(2xe^(sinx)+1))` `"Put "2xe^(sinx)+1=t^(2)` `rArr" "2[e^(sinx)+e^(sinx)cosx.x]dx=2tdt` `rArr" "e^(sinx)(1=x cos x)dx= tdt` `therefore" "l=int(tdt)/(((t^(2)-1))/(2)xt)` `l=2int(dt)/(t^(2)-1)=log|(t-1)/(t+1)|+C` where, `t^(2)=2xe^(sinx)+1` `l=log|(sqrt(2xe^(sinx)+1)-1)/(sqrt(2xe^(sinx)+1)-1)|+C` |
|
| 238. |
`int sin^(3) x cos^(3) x dx =?`A. `(1)/(4) sin^(4) x + (1)/(6) sin^(6) x+c`B. `(1)/(4) cos^(4)x-(1)/(6) cos^(6) x+c`C. none of the aboveD. |
| Answer» Correct Answer - A | |
| 239. |
`int 1/((x-1)^3(x+2)^5)^(1/4)dx` is equal toA. `(4)/(3)((x-1)/(x+2))^(1//4)+C`B. `(4)/(3)((x+2)/(x-1))^(1//4)+C`C. `(1)/(3)((x-1)/(x+2))^(1//4)+C`D. `(1)/(3)((x+2)/(x-1))^(1//4)+C` |
|
Answer» Correct Answer - C `int(1)/([(x-1)^(3)(x+2)^(5)]^(1//4))dx` `=int(1)/(((x-1)/(x+2))^(3//4)(x+2)^(2))dx` `=(1)/(3)int(1)/(t^(3//4))dt("put "(x-1)/(x+2)=t rArr (3)/((x+2)^(2))dx=dt)` `=(1)/(3)((t^(1//4))/((1)/(4)))+C=(4)/(3)t^(1//4)+C=(4)/(3)((x-1)/(x+2))^(1//4)+C` `=(1)/(3)((t^(1//4))/((1)/(4)))+C=(4)/(3)t^(1//4)+C=(4)/(3)((x-1)/(x+2))^(1//4)+C` |
|
| 240. |
`int(sin(x-alpha)/(sin(x+alpha)) dx`A. `x cos 2alpha +sin 2alpha log| sin (x+alpha)|+c`B. `x cos 2alpha -sin 2alpha log |sin (x+alpha)|+c`C. None of the aboveD. |
| Answer» Correct Answer - C | |
| 241. |
`intcos2thetalog((costheta+sintheta)/(costheta-sintheta))=`A. `(cos theta-sintheta)^(2)log((cos theta+sin theta)/(cos theta-cos theta))+C`B. `(cos theta+sin theta)^(2)log((cos theta+sin theta)/(cos theta-sin theta))+C`C. `((cos theta-sin theta)^(2))/(2)log((cos theta-sin theta)/(cos theta+sin theta))+C`D. `(1)/(2)sin 2 theta log tan ((pi)/(4+theta)-(1)/(2)log sec 2 theta +C` |
|
Answer» Correct Answer - D Since, `log((cos theta + sin theta)/(cos theta - sin theta))=log tan ((pi)/(4)+theta)` `"and "int sec theta d theta = log tan((pi)/(4)+(theta)/(2))` `rArr" "int sec 2 theta d theta =(1)/(2)log tan ((pi)/(4)+theta)` `" "2sec 2 theta=(d)/(d theta)log tan ((pi)/(4)+theta)` By integration by parts `therefore" "l=(1)/(2)sin 2 theta log tan ((pi)/(4)+theta)-int tan 2 theta d theta` `=(1)/(2)sin 2 theta log tan ((pi)/(4)+theta)-(1)/(2)log sec 2 theta +C` |
|
| 242. |
`int tan(x-alpha).tan(x+alpha).tan 2x dx` is equal toA. `ln|(sqrt(sec 2x).sec (x-alpha))/(sec(x-alpha))|+C`B. `ln|(sqrt(sec2x))/(sec(x-alpha).sec(x+alpha))|+C`C. `ln|(sqrt(sec2x).sec (x-alpha))/(sec(x+alpha))|+C`D. `ln|(sec 2x)/(sec(x-alpha).sec(x+alpha))|+C` |
|
Answer» Correct Answer - B `tan 2x=tan[(x=alpha)+(x+alpha)]` `=(tan(x-alpha)+tan(x+alpha))/(1-tan (x-alpha).tan(x+alpha))` `rArr" "tan(x-alpha).tan(x+alpha). tan 2x = tan 2x - tan (x-alpha)-tan(x+alpha)` `rArr tan(x-alpha).tan(x-alpha)-tan(x+alpha)]dx` `=(1)/(2)ln|sec 2x|-ln|sec(x-alpha)|-ln |sec(x+alpha)|+C` `=ln|(sqrt(sec2x))/(sec(x-alpha).sec(x+alpha))|+C` |
|
| 243. |
`int[(sqrt(x^2+1)(ln(x^2+1)-2lnx)))/(x^4) dx`A. `((x^(2)+1)^(3//2))/(x^(3))[(2)/(3)-ln((x^(2)+1)/(x^(2)))]+C`B. `((x^(2)+1)^(3//2))/(3x^(3))[ln((x^(2)+1)/(x^(2)))-(2)/(3)]+C`C. `((x^(2)+1)^(3//2))/(3x^(3))[(2)/(3)-ln((x^(2)+1)/(x^(2)))]+C`D. `(sqrt(x^(2)+1))/(3x^(3))(ln.(x^(2)+1)/(x^(2))-(2)/(3))+C` |
|
Answer» Correct Answer - C `I=int(sqrt(x^(2)+1).[ln(x^(2)+1)-2 In x])/(x^(4))dx` `=int sqrt(1+(1)/(x^(2))).(1)/(x^(3)).ln (1+(1)/(x^(2)))dx =-(1)/(2)int sqrt(t)` ln t dt on applying integration by parts, we get `=-(1)/(2)[ln t.(2t^(3//2))/(3)-int (1)/(t).(2)/(3)t^(3//2)dt]` `=-(1)/(3)t^(3//2)ln t +(2)/(9)t^(3//2)+C` `=((x^(2)+1)^(3//2))/(3x^(3)){(2)/(3)- ln((x^(2)+1)/(x^(2)))}+C` |
|
| 244. |
`int(sin 2x)/(5-cos^(2) x)dx` |
| Answer» Correct Answer - `log (5 -cos^(2) x) +c` | |
| 245. |
`(i) int (cos x-x sin x)/(x cos x) dx " "(ii) int(1+ cos x)/(x +sin x)^3 dx` |
| Answer» Correct Answer - `(i) log |x cos x| +c " "(ii) (1)/(-2(x+sin x)^(2)) +c` | |
| 246. |
`int(x^(e-1)-e^(x-1))/(x^e-e^x)dx` |
| Answer» Correct Answer - `(1)/(e)log |x^(e)-e^(x)|+c` | |
| 247. |
`int e^(x). " sin x( sin x+ 2 cos x) dx "` |
| Answer» Correct Answer - `e^(x) . Sin^(2) x+c` | |
| 248. |
`int(tan(1+log x))/(x) dx` |
| Answer» Correct Answer - `1 log |sec (1+ log x)|+c` | |
| 249. |
`(i) int e^(x). "[log (sec x+tan x) + sec x dx "` `(ii) int (e^(-x)(cos x-sin x))/(cos^(2) x) dx` |
|
Answer» Correct Answer - `(i) e^(x),log (sec x + tan x) +c` `(ii) -e^(-x) sec x +c` |
|
| 250. |
`int e^x (logsin x + cot x)dx` |
| Answer» Correct Answer - `e^(x) , log (sin x) +c` | |