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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.
| 251. |
integrate `int_0^(2pi) e^x . sin (pi/4 + x/2) dx` |
| Answer» Correct Answer - B | |
| 252. |
Integrate the functions`1/(cos(x-a)cos(x-b)`A. `tan x+ cosec x+c`B. `-tan x+ cot x+c`C. `tan x + sec x+c`D. |
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Answer» Correct Answer - a ` int (sin^(2) x-cos ^(2)x)/(sin^(2) x cos^(2) x)` ` = int ((sin^(2)x)/(sin^(2) x cos^(2) x)-(cos^(2)x)/(sin^(2) x cos^(2)x))dx` `=int ((1)/(cos^(2)x)-(1)/(sin^(2)x))dx` ` = int (sec^(2) x- " cosec"^(2) x) dx` `= tan x + cot x+c` |
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| 253. |
`int(x^3+3)/(x^3-3x)dx` |
| Answer» Correct Answer - `x+(sqrt(3)+1)/(2) log |x-sqrt(3) |-(sqrt(3)-1)/(2)log |x-sqrt(3) |-log |x|+c` | |
| 254. |
Find `int(x^2)/((x^2+1)(x^2+4)dx` |
| Answer» Correct Answer - `-(1)/(5) tan^(-1) x+(2)/(3)tan^(-1).(x)/(2) +c` | |
| 255. |
`int sin^(3)x cos^(3)x dx ` is equal toA. `(1)/(32)[-(3)/(2)cos 2x+(1)/(6)cos 6x]+C`B. `(1)/(16)[-(3)/(2)cos 2x+(1)/(6)cos 6x]+C`C. `(1)/(64)[-(3)/(2)cos 2x +(1)/(6)cos 6x]+C`D. None of the above |
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Answer» Correct Answer - A `intsin^(3)x cos^(3)x dx=(1)/(8)int(2sin x cosx)^(3)dx` `=(1)/(8)intsin^(3)2xdx` `=(1)/(8)int(3sin2x-sin6x)/(4)dx` `=(1)/(32)int(3sin2x-sin6x)dx` `=(1)/(32)[-(3)/(2)cos2x+(1)/(6)cos 6x]+C` |
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| 256. |
`int(x^2)/(x^6+x^3-2)dx` |
| Answer» Correct Answer - `(1)/(9) .log |(x^(3)-1)/(x^(3)+2)|+c` | |
| 257. |
`int(x^2)/((x-1)(x-2)(x-3))dx` |
| Answer» Correct Answer - `-(1)/(6) log |x+1|+(4)/(15) log |x+2|+(9)/(10)log |x+3|+c` | |
| 258. |
The anti derivative of `(sqrt(x)+1/(sqrt(x)))`equals(A) `1/3x^(1/3)+2x^(1/2)+C` (B) `2/3x^(2/3)+1/2x^2+C`(C) `2/3x^(3/2)+2x^(1/2)+C` (D) `3/2x^(3/2)+1/2x^(1/2)+C`A. `(2)/(3)x^(2/3) +(1)/(2)x^(2)+c`B. `(2)/(3)x^(3/2) +2x^(1/2)+c`C. `(3)/(2) x^(3/2)+(1)/(2)x^(1/2)+c`D. |
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Answer» Correct Answer - c `int (sqrt(x) +(1)/(sqrt(x)))dx =int x^(1//2) dx+ int x^(-1//2) dx` `= (x^(3//2))/(3//2) +(x^(1//2))/(1//2) +c =(2)/(3)x^(3//2) + 2x^(1//2) +c` |
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| 259. |
`int (e^(x)dx)/(e^(2x)+4e^(x)+3)` |
| Answer» Correct Answer - `(1)/(2) log |(1+e^(x))/(3+e^(x))|+c` | |
| 260. |
The anti-derivative of `sin 2x-4e^(3x)` isA. `(1)/(2)cos 2x-(4)/(3)e^(3x)+C`B. `-(1)/(2)cos 2x+(4)/(3)e^(3x)+C`C. `-(1)/(2)cos 2x-(4)/(3)e^(3x)+C`D. None of these |
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Answer» Correct Answer - C We have, `int(sin 2x-4e^(3x))dx` `=intsin 2xdx-4inte^(3x)dx` `=(-cos2x)/(2)-4e^(3x)dx` `=(-cos2x)/(2)-4(e^(3x))/(3)+C` `=-(1)/(2)cos2x-(4)/(3)e^(3x)+C` |
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| 261. |
`int(dx)/(sqrt(1+4x^2))`A. `(1)/(2)log|x+sqrt(2x^(2)+1)|+C`B. `(1)/(2)log|2x+sqrt(4x^(2)+1)|+C`C. `log|2x+sqrt(4x^(2)+1)|+C`D. None of these |
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Answer» Correct Answer - B `int(dx)/(sqrt(1-4x^(2)))=int(dx)/(sqrt(4{((1)/(2))^(2)+x^(2)}))` `=(1)/(2)int(dx)/(sqrt(x^(2)+((1)/(2))^(2)))=(1)/(2)log|x+sqrt(x^(2)+((1)/(2)))^(2)|+C_(1)` `=(1)/(2)log|x+(sqrt(4x^(2)+1))/(2)|+C_(1)` `=(1)/(2)log|2x+sqrt(4x^(2)+1)|-(1)/(2)log2+C_(1)` `=(1)/(2)log|2x+sqrt(4x^(2)+1)|+C" "[C=(1)/(2)log2+C_(1)]` |
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| 262. |
`int(1/sqrt(9-25x^2))` dxA. `sin^(-1)((5x)/(3))+C`B. `(3)/(2)sin^(-1)((5x)/(3))+C`C. `(1)/(5)sin^(-1)((5x)/(3))+C`D. None of these |
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Answer» Correct Answer - C `int(dx)/(sqrt(9-25x^(2)))=int(dx)/(sqrt(25[((3)/(5))^(2)-x^(2)]))=(1)/(5)int(dx)/(sqrt([((3)/(5))^(2)-x^(2)]))` `=(1)/(5)sin^(-1)((x)/(3//5))+C` `=(1)/(5)sin^(-1)((5x)/(3))+C` |
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| 263. |
`int(1)/(sin^(2)x.cos^(2)x)dx` is equal toA. `sinx - cos x +C`B. `tanx +cot x+C`C. `cos x +sinx +C`D. `tanx -cot x +C` |
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Answer» Correct Answer - D `int(1)/(sin^(2)xcos^(2)x)dx=int(sin^(2)x+cos^(2)x)/(sin^(2)xcos^(2)x)dx` `=int((1)/(cos^(2)x)+(1)/(sin^(2)x))dx` `=int(sec^(2)x+"cosec"^(2)x)dx=tanx-cot x+C` |
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| 264. |
`int(root3(x))(root5(1+root3(x^(4))))dx` is equal toA. `(1+x^((3)/(4)))^((6)/(5))+C`B. `(1+x^((4)/(3)))^((6)/(5))+C`C. `(5)/(8)(1+x^((4)/(3)))^((6)/(5))+C`D. `(1)/(6)(1+x^((4)/(3)))^(6)+C` |
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Answer» Correct Answer - C Let `l=int(root3(x))(root5(1+root3(x^(4))))dx` Put `root3(x)^(4)=t rArr (4)/(3).root3x dx=dt` `therefore" "l=(3)/(4)int(root5(1+t))dt=(3)/(4)[((1+t)^((1)/(5)+1))/((1)/(5)+1)]+C` `=(5)/(8)[(1+root3(x)^(4))^(6//5)]+C` |
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| 265. |
`int(cos^(2) (log .x))/(x)dx` |
| Answer» Correct Answer - `(1)/(2) log x+(1)/(4) sin(2 log x) +c` | |
| 266. |
`int(dx)/(a^(2)sin^(2)x+b^(2)cos^(2)x)` is equal toA. `(1)/(ab)tan^(-1)((a tanx)/(b))+C`B. `(1)/(b)tan^(-1)((atanx)/(b))+C`C. `(b)/(a)tan^(-1)((btanx)/(a))+C`D. None of these |
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Answer» Correct Answer - A `int(dx)/(a^(2)sin^(2)x+b^(2)cos^(2)x)` `=int(sec^(2)xdx)/(a^(2)tan^(2)x+b^(2))=(1)/(a^(2))int(dt)/(t^(2)+((b)/(a))^(2))` where, t = tan x `=(1)/(a^(2)).(a)/(b)tan^(-1)(t)/(b//a)+C=(1)/(ab).tan^(-1){(a(tanx))/(b)}+C` |
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| 267. |
`inte^(e^(e^(x))).e^(e^(x)).e^(x)dx=....+C`A. `e^(e^(x))`B. `(1)/(2)e^(x).e^(x)`C. `e^(e^(e^(x)))`D. `(1)/(2)e^(e^(x))` |
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Answer» Correct Answer - C Let `l=inte^(e^(e^(x)))e^(e^(x))e^(x)dx` `"Put "e^(e^(e^(x)))=trArr e^(e^(e^(x)))e^(e^(x))e^(x)dx=dt` `therefore" "l=intdt=t=e^(e^(e^(x)))` |
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| 268. |
`(i) int " x sec"^(2) " 2x dx "" "(ii) int " x sin"^(3) " x dx "` |
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Answer» Correct Answer - `(i) (1)/(2) xtan 2x-(1)/(4) log| sec 2x |+c` `(ii) -(3)/(4) x cos x +(3)/(4) sin x +(1)/(12) x cos 3x -(1)/(36) sin 3x+c` |
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| 269. |
`int(1)/(x cos^(2)(log_(e)x))dx` |
| Answer» Correct Answer - `tan (log_(e) x) +c` | |
| 270. |
Evaluate `int sin^(-1)x dx`. |
| Answer» Correct Answer - `x sin^(-1) x+sqrt(1-x^(2))+c` | |
| 271. |
`int(cot x)/(1+sinx) dx` |
| Answer» Correct Answer - `log|1+sinx|+c` | |
| 272. |
`int(1)/(x)(log_(ex)e)dx` is equal toA. `log_(e)(1-log_(e)x)+C`B. `log_(e)(log_(e)ex-1)+C`C. `log_(e)(log_(e)x-1)+C`D. `log_(e)(log_(e)x+1)+C` |
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Answer» Correct Answer - D Let `l=int(1)/(x)(log_(ex)e)dx=int(1)/(x(1+log_(e)x))dx` Put `log_(e)x=t" "rArr" "(1)/(x)dx=dt` `therefore" "l=int(dt)/((1+t))=log_(e)(1+t)+C` `=log_(e)(1+log_(e)x)+C` |
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| 273. |
`int cot^(-1)x dx` |
| Answer» Correct Answer - `x cot^(-1) x+(1)/(2) log (1+x^(2)) +c` | |
| 274. |
`int(sin^(-1)x)/(sqrt(1-x^(2)))dx` is equal to Where, C is an arbitrary constant.A. `log(sin^(-1)x)+C`B. `(1)/(2)(sin^(-1)x)^(2)+C`C. `log(sqrt(1-x^(2)))+C`D. `sin(cos^(-1)x)+C` |
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Answer» Correct Answer - B Let `" "l=int(sin^(-1)x)/(sqrt(1-x^(2)))dx` Put`" "sin^(-1)x=t" "rArr" "(1)/(sqrt(1-x^(2)))dx=dt` `therefore" "l=int t dt=(t^(2))/(2)+C=((sin^(-1)x)^(2))/(2)+C` |
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| 275. |
`int((a+bsin^(-1)x)^(n))/(sqrt(1-x^(2)))dx` |
| Answer» Correct Answer - `((a+b sin^(-1) x)^(n+1))/(b(n+1)) +c` | |
| 276. |
`int x^3 e^(x^2) dx` is equal to |
| Answer» Correct Answer - `(1)/(2) e^(x^(2)) (x^(2)-1) +c` | |
| 277. |
`int(dx)/((x+1)sqrt(4x+3))` is equal toA. `tan^(-1)sqrt(4x+3)+C`B. `3tan^(-1)sqrt(4x+3)+C`C. `2tan^(-1)sqrt(4x+3)+C`D. `4tan^(-1)sqrt(4x+3)+C` |
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Answer» Correct Answer - C Let `l=int(dx)/((x+1)sqrt(4x+3))` Put `4x+3=t^(2) rArr 4 dx = 2t dt` `therefore" "l=(1)/(2)int(tdt)/(((t^(2)-3)/(4)+1)t)=2int(dt)/(1+t^(2))` `=2tan^(-1)t+C=2tan^(-1)sqrt(4x+3)+C` |
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| 278. |
`intsin^(- 1)((2x)/(1+x^2))dx` |
| Answer» Correct Answer - `2[ xtan^(-1) x-(1)/(2)log (1+x^(2))]+c` | |
| 279. |
`intsin^(5) x dx` |
| Answer» Correct Answer - `-cos x-(1)/(5)cos^(5)x+(2)/(3) cos^(3) x+c` | |
| 280. |
`intsin^(2) n x dx` |
| Answer» Correct Answer - `(1)/(2) x -(1)/(4n) sin 2nx +c` | |
| 281. |
`intsin^(3)x.cos^(2)xdx` is equal toA. `(sin^(5)x)/(5)-(sin^(3)x)/(3)+C`B. `(sin^(5)x)/(5)+(sin^(3)x)/(3)+C`C. `(cos^(5)x)/(5)-(cso^(3)x)/(3)+C`D. `(cos^(5)x)/(5)+(cos^(3)x)/(3)+C` |
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Answer» Correct Answer - C Let `l=int(1-cos^(2)x) cos^(2)x. sinx x` Put `cos x = t rArr - sin x dx=dt` `therefore" "l=- int(1-t^(2))t^(2)dt=int(t^(4)-t^(2))dt` `=(t^(5))/(5)-(t^(3))/(3)+C` `=((cosx)^(5))/(5)-((cosx)^(3))/(3)+C` |
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| 282. |
`int(sinxcosx)/(sqrt(1-sin^(4)x)dx` is equal toA. `(1)/(2)sin^(-1)(sin^(2)x)+C`B. `(1)/(2)cos^(-1)(sin^(2)x)+C`C. `tan^(-1)(sin^(2)x)+C`D. `tan^(-1)(2sin^(2)x)+C` |
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Answer» Correct Answer - A Let `l=int(sinx cos x)/(sqrt(1-sin^(4)x))dx` Put`" "Sin^(2)x=t rArr 2 sin x cos x dx=dt` `therefore" "l=int(dt)/(2sqrt(1-t^(2)))=(1)/(2)sin^(-1)t+C` `=(1)/(2)sin^(-1)(sin^(2)x)+C` |
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| 283. |
If `int(sqrt(cotx))/(sinxcosx)dx=Psqrt(cotx)+Q,` then the value of P isA. `-2`B. 2C. 3D. `-3` |
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Answer» Correct Answer - A Let `l=int(sqrt(cotx))/(sinx cosx)dx=int("cosec"^(2)x sqrt(cotx)dx)/(cotx)` `" "["divide numerator and denominator by sin"^(2)x]` `=int("cosec"^(2)x)/(sqrt(cotx))dx` Put `cos x=t rArr -"cosec"^(2)x dx = dt` `therefore" "l=int(-dt)/(sqrtt)=(-t^(1//2))/(1//2)+C=-2sqrt(cotx)+C` `therefore" "P=-2` |
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| 284. |
`intsqrt(2x+(1)/(3) dx)` |
| Answer» Correct Answer - `(1)/(3)(2x+(1)/(3))^(3//2)+c` | |
| 285. |
`int(1)/((a+bx)^(5) )dx` |
| Answer» Correct Answer - `(1)/(-4b (a+bx)^(4))+c` | |
| 286. |
`int(1)/((7x-2)^(2))dx` |
| Answer» Correct Answer - `-(1)/(7(7x-2))+c` | |
| 287. |
`int sqrt(2x-1) dx` |
| Answer» Correct Answer - `((2x-1)^(3//2))/(3)+c` | |
| 288. |
`int(3 -7x)^(5) dx` |
| Answer» Correct Answer - `((3-7x)^(6))/(-42)+c` | |
| 289. |
Evaluate: `int(a x+b)^3dx` |
| Answer» Correct Answer - `((ax +b)^(4))/(4a) +c` | |
| 290. |
Evaluate: `intxlog(1+x) dx` |
| Answer» Correct Answer - `(x^(2)-1)/(2) log (1+x) -(x^(2))/(4)+(x)/(4) +c` | |
| 291. |
`int(1-x)/(sqrt(x))dx` |
| Answer» Correct Answer - `2sqrt(x)-(2)/(3)x^(3//2) +c`. | |
| 292. |
`int(sin^(- 1)x)/((1-x^2)^(3/2))dx` |
| Answer» Correct Answer - `(x sin^(-1)x)/(sqrt(1+x^(2)))+(1)/(2) log (1-x^(2)) +c` | |
| 293. |
`int " x sec"^(2) " x dx "` |
| Answer» Correct Answer - `x tan x- log |secx|+c` | |
| 294. |
`int(10 x^9+10 x^x(log)_(e^(10))dx)/(x^(10)+10^x)` equalsA. `10^(x)-x^(10)+c`B. `10^(x) +x^(10)+c`C. `(10^(x) -x^(10))^(-1)+c`D. `log (10^(x) +x^(10))+c` |
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Answer» Correct Answer - D `int [(10 x^(9) +10^(x)log _(e) 10)/(x^(10)+10^(x))]dx` ` " Let " x^(10) +10^(x) =t rArr (10x^(9) +10^(x) log_(e) 10) dx=dt` `=int (1)/(t) dt= log | t| +c` ` = log |x^(10)+ 10^(x)| +c` lt brgt |
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| 295. |
`(i) int ((1+x)^(3))/(sqrt(x))dx " "(ii) int((1+x)^(3))/(x^(4)) dx` |
| Answer» Correct Answer - `(i) (2)/(7) x^(7//2) +(6)/(5) x^(5//2) +2x^(3//2) +2sqrt(x)+c " "(ii) -(1)/(3x^(3))-(3)/(2x^(2)) -(3)/(x)+ log x+c` | |
| 296. |
`int(x^2tan^(- 1)x)/(1+x^2)dx` |
| Answer» Correct Answer - `x tan^(-1) x-(1)/(2) log (1+x^(2)) -(1)/(2) (tan^(-1)x)^(2) +c` | |
| 297. |
`int sinx/sin(3x) dx=` |
| Answer» Correct Answer - `(1)/(2sqrt(3)) log |(sqrt(3)+tanx)/(sqrt(3)-tan x)|+c` | |
| 298. |
`int x. a^(x) dx ` |
| Answer» Correct Answer - `(a^(x))/(log a) [x-(1)/(log a) ]+c` | |
| 299. |
Evaluate:`int(10x^9+10^x(log)_e 10)/(10^x+x^(10))dx`A. `10^(x)-x^(10)+C`B. `10^(x)+x^(10)+C`C. `10^(x)-(x^(10)-x^(-1))+C`D. `log|10^(x)+x^(10)|+C` |
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Answer» Correct Answer - D `int((10x^(9)+10^(x)log_(e)10)/(x^(10)+10^(x)))dx` Let`" "x^(10)+10^(x)=t` `rArr (10x^(9)+10^(x)log_(e)10)dx=dt` `=int(dt)/(t)=log|t|+Clog|10^(x)+x^(10)|+C` |
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| 300. |
`int(e^(x)+cos x)/(e^(x) +sin x) dx` |
| Answer» Correct Answer - `log|e^(x)+sin x| +c` | |