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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.
| 351. |
Evaluate :`int_0^1x(1-x)^5dx` |
| Answer» Correct Answer - `(1)/(42)` | |
| 352. |
`int(1)/(sin x cos x + 2cos^(2)x)dx` |
| Answer» Correct Answer - `log| tan x+2|+c` | |
| 353. |
`(i) int(2cos x)/(sqrt(1-4cos^(2) x))dx " "(ii) int (x+1)/(sqrt(x^(2)+1))dx` |
| Answer» Correct Answer - `(8) log "|" sin x+ sqrt (sin^(2) x-(3)/(4))|+c" "(ii) sqrt(x^(2) +1)+ log | x+ sqrt(x^(2) +1)"|"+c` | |
| 354. |
Evaluate: (i) `int1/(sqrt(1+cos2x)) dx`(ii) `int1/(sqrt(1-cosx)) dx` |
| Answer» Correct Answer - `sqrt(2) log |tan.(x)/(4)|+c` | |
| 355. |
`int1/(sqrt(8+3x-n^2))dx`A. `(2)/(3)sin^(-1)((2x-1)/(sqrt(41)))+C`B. `(3)/(2)sin^(-1)((2x-3)/(sqrt(41)))+C`C. `(1)/(sqrt(41))sin^(-1)((2x-3)/(sqrt(41)))+C`D. `sin^(-1)((2x-3)/(sqrt(41)))+C` |
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Answer» Correct Answer - D Let `I=int(1)/(sqrt(8+3x-x^(2)))dx` `=int(1)/(sqrt(8-[x^(2)-3x+((3)/(2))^(2)-((3)/(2))^(2)]))dx` `=int(1)/(sqrt(8-[(x-(3)/(2))^(2)-(9)/(4)]))dx` `=int(1)/(sqrt(8+(9)/(4)-(x-(3)/(2))^(2)))dx` `=int(1)/(sqrt(((sqrt(41))/(2))^(2)-(x-(3)/(2))^(2)))dx` Let `x-(3)/(2)=t rArr dx=dt` `therefore" "I=int(1)/(sqrt(((sqrt(41))/2)^(2)-t^(2)))dt=sin^(-1)((t)/((sqrt(41))/(2)))+C` `=sin^(-1)((2x-3)/(sqrt(41)))+C` |
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| 356. |
`int cos (2x +1) dx` |
| Answer» Correct Answer - `(1)/(2) sin (2x +1) +c` | |
| 357. |
`intsec^(6//5) x. "cosec"^(4//5) x dx` |
| Answer» Correct Answer - `5(tanx)^(1//5) +c` | |
| 358. |
Evaluate: `int1/(sqrt(4x^2-9)) dx` |
| Answer» Correct Answer - `(1)/(2) sin ^(-1) .(2x)/(5) +c` | |
| 359. |
`intsec^(2) (1-5x) dx` |
| Answer» Correct Answer - `-(1)/(5) tan (1-5x) +c` | |
| 360. |
Evaluate: `int1/(cos(x-a)cos(x-b)) dx` |
| Answer» Correct Answer - `(1)/(sin(a-b)) log |(cos (x-a))/(cos(x-b))| +c` | |
| 361. |
Evaluate :`int1/(1+cos tx)dx` |
| Answer» Correct Answer - `" cosec x- cot x+c "` | |
| 362. |
`int1/(cos^2x(1-tanx)^2) dx` |
| Answer» Correct Answer - `(1)/(1-tanx)+c` | |
| 363. |
`int(x dx)/((1+x^2)^(3//2))` |
| Answer» Correct Answer - `-(1)/(sqrt(1+x^(2)))+c` | |
| 364. |
`int_(0)^(a) (x)/(sqrt(a^(2)-x^(2)))dx` |
| Answer» Correct Answer - `a` | |
| 365. |
`int_(0)^(pi//2) sin^(2) x cos ^(2) x dx` |
| Answer» Correct Answer - `(2)/(15)` | |
| 366. |
Evaluate: (i) `int(sinx)/(1+cos^2x) dx`(ii) `int(2x^3)/(4+x^8) dx` |
| Answer» Correct Answer - `(i) (1)/(4)tan^(-1) .(x^(4))/(2)+c (ii) cos ((1)/(x)) +c ` | |
| 367. |
Evaluate the integrals`int0pi/2(sinx)/(1+cos^2x)dx` |
| Answer» Correct Answer - `(pi)/(4)` | |
| 368. |
`inte^(4-3x) dx ` |
| Answer» Correct Answer - `-(1)/(3).e^(4-3x) +c` | |
| 369. |
Evaluate: `int(1+x+x^2)/(x^2(1+x)) dx` |
| Answer» Correct Answer - `-(1)/(x) +log_(e) (1+x) +c` | |
| 370. |
`intz^(-1//3) " dz "` |
| Answer» Correct Answer - `(3)/(2) z^(2//3)+c` | |
| 371. |
`int1/(x(x^n+1))dx`A. `(1)/(n) log {x^(n) (x^(n) +1))} +c`B. `log ((x^(n))/(x^(n)+1)))+c`C. None of the aboveD. |
| Answer» Correct Answer - A | |
| 372. |
Find`int(sin^6x)/(cos^8x)dxdot`A. `(tan^(6)x)/(6)+C`B. `(tan^(7)x)/(7)+C`C. `(tan^(8)x)/(8)+C`D. None of these |
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Answer» Correct Answer - B We have,`" "l=int(sin^(6)x)/(cos^(8)x)dx` `l=int((sinx)/(cosx))^(6)((1)/(cos^(2)x))dx` `l=int tan^(6)x sec^(2)xdx` Put `tanx=t rArr sec^(2) x dx = dt` `therefore" "l=int t^(6)dt rArr l=(t^(7))/(7)+C` `l=(tan^(7)x)/(7)+C` |
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| 373. |
`inte^(-logx)dx` is equal toA. `e^(-logx)+C`B. `-xe^(-logx)+C`C. `e^(logx)+C`D. `log|x|+C` |
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Answer» Correct Answer - D Let `l=inte^(-logx)dx=int(1)/(x)dx=log|x|+C` |
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| 374. |
`int{ 5a^(x)+ 6a cos (5x +1)}dx` |
| Answer» Correct Answer - `(5.a^(x))/(log a) + (6a)/(5) sin (5x+1) +c` | |
| 375. |
`int(1)/(sqrt(1-(3x+2)^(2)))dx` |
| Answer» Correct Answer - `(1)/(3)sin^(-1) (3x+2) +c` | |
| 376. |
`int (1)/(sqrt(x)) dx` |
| Answer» Correct Answer - `2sqrt(x)+c` | |
| 377. |
`int x^(-6) " dx "` |
| Answer» Correct Answer - `-(1)/(5x^(5)) +c` | |
| 378. |
Evaluate:`inta/(b+c e^x)dx` |
| Answer» Correct Answer - `-(a)/(b).log |b.e^(-x) +c|+c_(1)` | |
| 379. |
If`int1/(x+x^5)dx=f(x)+c ,t h e ne v a l u a t eint(x^4)/(x+x^5)dxdot`A. `logx-f(x)+C`B. `f(x)+logx+C`C. `f(x)-logx+C`D. None of these |
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Answer» Correct Answer - A Let`" "l=int(x^(4))/(x+x^(5))dx=int(x^(4)+1-1)/(x+x^(5))dx` `rArr" "l=int(1)/(x)dx-int(1)/(x^(5)+x)dx` `rArr" "l=logx-[f(x)+C]" [Given]"` `rArr" "l=logx-f(x)+C_(1)" "["where, "C_(1)=-C]` |
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| 380. |
`int(x+2)sqrt(x^(2)+x)dx` |
| Answer» Correct Answer - `(1)/(3) (x^(2) +x+1)^(3//2) +(3)/(8) (2x+1)sqrt(x^(2)+x+1)+(9)/(16)log |x+(1)/(2)+sqrt(x^(2)+x+1)|+c` | |
| 381. |
` (i) int sqrt(1+ sin .(x)/(2)dx)` `(ii) int (1+cos 4x)/(cot x- tan x)dx` |
| Answer» Correct Answer - `(i) 4 sin .(x)/(4) -4 cos .(x)/(4)+c " "(ii) -(1)/(8) cos4x +c` | |
| 382. |
`int(1dx)/((x-2)sqrt(x^(2)-4x+3) ` |
| Answer» Correct Answer - `sec^(-1) (x-2) +c` | |
| 383. |
`int xsqrt(x^(4)+9) dx` |
| Answer» Correct Answer - `(x^(2))/(4) sqrt(x^(4) +9) +(9)/(4) log | x^(2) +sqrt(x^(4)+9)|+c` | |
| 384. |
`int(dx)/(cos^(3)xsqrt(2sin 2x))` is equal toA. `sqrt(tanx)+(tan^(5//2)x)/(5)+C`B. `sqrt(tanx)+(2)/(5)tan^(5//2)x+C`C. `2sqrt(tanx)+(2)/(5)tan^(5//2)x+C`D. None of the above |
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Answer» Correct Answer - A Let `l=int(dx)/(cos^(3)x sqrt(2sin 2x))=int(dx)/(cos^(3)x sqrt(4sin x cos x))` `=(1)/(2)int(dx)/(cos^(7//2)x sin^(1//2)x)` `=(1)/(4)int(sec^(4)x)/(sqrt(tanx))dx` `=(1)/(2)int((1+tan)^(2)sec^(2)x)/(sqrt(tanx))` Put `tan x = t rArr sec^(2)x dx =dt` `therefore" "l=(1)/(2)int(1+t^(2))/(sqrtt)dt` `=(1)/(2)int t^(-1//2)dt+(1)/(2)intt^(3//2)dt=intt+(t^(5//2))/(5)+C` `=sqrt(tanx)+(1)/(5)tan^(5//2)x+C` |
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| 385. |
`int(1)/(xsqrt(1-x^(3)))`dx is equal toA. `(1)/(3)log((1)/(sqrt(1-x^(3))))+C`B. `(2)/(3)log((1)/(sqrt(1-x^(3))))+C`C. `(1)/(3)log((sqrt(1-x^(3))-3)/(sqrt(1-x^(3)+3)))+C`D. `(1)/(3)log((sqrt(1-x^(3))-3)/(sqrt(1-x^(3))+1))+C` |
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Answer» Correct Answer - D `l=(dx)/(xsqrt(1-x)^(3))` On multiplying numerator and denominator by `x^(2)`, we get `l=int(x^(2)dx)/(x^(3)sqrt(1-x^(3)))` On putting `1-x^(3)=t^(2)=t = sqrt(1-x^(3))` `rArr" "-3x^(2)dx=2t dt` `rArr" "x^(2)dx=-(2)/(3)t dt and x^(3)=1-t^(2)` `therefore" "l=-(2)/(3)int(tdt)/((1-t^(2))t)` `=(2)/(3)int(dt)/(t^(2)-1)=(2)/(3)[(1)/(2)log|(t-1)/(t+1)|]+C` `=(1)/(3)log|(t-1)/(t+1)|+C` `=(1)/(3)log|(sqrt(1-x^(3))-1)/(sqrt(1-x^(3))+1|+C` |
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| 386. |
`int (1)/(xsqrt(x))dx` |
| Answer» Correct Answer - `-2.x^(-1//2) +c` | |
| 387. |
`" if " f(x) ={ underset( 5x" "2 le x le 3) (3x+ 4,0 le x le 2),` then `int_(0)^(3) f(x) dx=?`A. `(53)/(2)`B. `(55)/(2)`C. `(57)/(2)`D. |
| Answer» Correct Answer - B | |
| 388. |
Evaluate: `int(x-5) sqrt(x^2+x) dx` |
| Answer» Correct Answer - `(1)/(3) (x^(2) -x-2)^(3//2) -(11)/(8)(2x +1)sqrt(x^(2)-x-2)-(11)/(16) log |x-(1)/(2)+sqrt(x^(2)-x-2)|+c` | |
| 389. |
Evaluate: `int(2x+5)/(sqrt(x^2+2x+5)) dx` |
| Answer» Correct Answer - `2sqrt(x^(2)+2x+5)+3log |x+1+sqrt(x^(2)+2x+5)|+c` | |
| 390. |
`int(1)/(1+3 cos^(2) x)dx` |
| Answer» Correct Answer - `(1)/(2) tan^(-1) ((tanx)/(2))+c` | |
| 391. |
Evaluate: `int(4x+1) sqrt(x^2-x-2) dx` |
| Answer» Correct Answer - `(4)/(3) (x^(2) -x-2)^(3//2) +(3)/(4)(2x-1)sqrt(x^(2)-x-2)-(27)/(8) log |x-(1)/(2)+sqrt(x^(2)-x-2)|+c` | |
| 392. |
`int(dx)/(xsqrt(x^(6)-16))=`A. `(1)/(3)sec^(-1)((x^(3))/(4))+C`B. `cos^(-1)((x^(3))/(4))+C`C. `(1)/(12)sec^(-1)((x^(3))/(4))+C`D. `sec^(-1)((x^(3))/(4))+C` |
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Answer» Correct Answer - C Let `l=int(dx)/(xsqrt(x^(6)-16))=(1)/(3)int(3x^(2))/(x^(3)sqrt((x^(3))^(2)-4^(2)))dx` Put `x^(3)=t rArr 3x^(2)dx=dt` `therefore" "l=(1)/(3)int(dt)/(sqrt(t^(2)-4^(2)))=(1)/(3xx4)sec^(-1)((t)/(4))+C` `=(1)/(12)sec^(-1)((x^(3))/(4))+C` |
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| 393. |
`" if " int (sin 2x- cos 2x) dx=(1)/(sqrt(2)) sin (2x-k)+c " then " k=?`A. `-(5pi)/(4)`B. `(pi)/(4)`C. `-(pi)/(4)`D. |
| Answer» Correct Answer - A | |
| 394. |
`intsqrt(cotx) dx` |
| Answer» Correct Answer - `-(1)/(sqrt(2))tan^(-1) ((cot x-1)/(sqrt(2)cot x))- (1)/(2sqrt(2)) log |(cot x+1-sqrt(2cot x))/(cot x+1+sqrt(2 cot x))|+c` | |
| 395. |
`int (1-cosx) /(cos x(1+cos x)) dx` |
| Answer» Correct Answer - `log | sec x+ tan x| -2 tan .(x)/(2) +c` | |
| 396. |
`int(1)/(4+5 cosx)dx` |
| Answer» Correct Answer - `(1)/(3) log |(3+tan .(x)/(2))/(3-tan.(x)/(2))|+c` | |
| 397. |
`int (cosx)/(cos 3x)dx` |
| Answer» Correct Answer - `(1)/(2sqrt(3))log |(1+sqrt(3)tanx)/(1-sqrt(3) tanx)|+c` | |
| 398. |
`int(cosx)/((1+sinx)(2+sinx))dx` |
| Answer» Correct Answer - `-(1)/(3) log|x+1|+(1)/(4)log|x-2|+(1)/(12)log |x+2|+c` | |
| 399. |
Evaluate:`int(2x-5)sqrt(2+3x-x^2)dx` |
| Answer» Correct Answer - `-(2)/(3) (2+3x-x^(2))^(3//2) -(1)/(2) (2x-3)sqrt(2+3x-x^(2))-(17)/(4)sin^(-1) ((2x-3)/(sqrt(17))) +c` | |
| 400. |
`int(sin x)/( sqrt(1+cos x))dx=?`A. `sqrt(1+cos x) +c`B. `-2sqrt(1+ cos x)+c`C. `2sqrt(1+ cos x) +c`D. |
| Answer» Correct Answer - C | |