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.
| 51. |
Evaluate the following integrals: `inte^(sqrt(x))dx` |
|
Answer» Correct Answer - `2e^(sqrt(x))(sqrt(x)-1)+C` Put `sqrt(x)=t anddx="2t dt"`. |
|
| 52. |
`intxsinxcosxdx=?`A. `-(1)/(4)xcos2x+(1)/(8)sin2x+C`B. `(1)/(4)xcos2x+(1)/(8)sin2x+C`C. `(1)/(4)xcos2x-(1)/(8)sin2x+C`D. none of these |
|
Answer» Correct Answer - A `I=(1)/(2)intunderset(I)(x)underset(II)(sin2x)dx`. |
|
| 53. |
`int(sinsqrt(x))/(sqrt(x))dx=?`A. `2cossqrt(x)+C`B. `-2cossqrt(x)+C`C. `-(cossqrt(x))/(2)+C`D. `(cossqrt(x))/(2)+C` |
|
Answer» Correct Answer - B Put `sqrt(x)=t and(1)/(sqrt(x))dx=2dt`. `:." "I=2intsindt=-2cost+C=-2cossqrt(x)+C`. |
|
| 54. |
Evaluate the following integrals: `int(dx)/((acosx+bsinx)^(2)),agt0andbgt0` |
|
Answer» Correct Answer - `(1)/((a^(2)+b^(2)))tan{x-tan^(-1)""(b)/(a)}+C` `Puta=rsinthetaandb=rcostheta` |
|
| 55. |
Evaluate the following integrals: `intcos^(-1)((1-x^(2))/(1+x^(2)))dx` |
|
Answer» Correct Answer - `2xtan^(-1)x-log(1+x^(2))+C` Put `x=tant anddx=sec^(2)t" dt. Use"((1-tan^(2)t)/(1+tan^(2)t))=cos2t`. |
|
| 56. |
Evaluate the following integrals: `intcos^(-1)(4x^(3)-3x)dx` |
|
Answer» Correct Answer - `3xcos^(-1)x-3sqrt(1-x^(2))+C` Put `x=cost anddx=-sin"t dt. Use"(4cos^(3)t-3cost)=cos3t`. |
|
| 57. |
Evaluate the following integrals: `intsin^(-1)sqrt(x)dx` |
|
Answer» Correct Answer - `(1)/(2)(2x-1)sin^(-1)sqrt(x)+(1)/(2)sqrt(x(1-x))+C` Put `sqrt(x)=sint."Then",x=sin^(2)t anddx=sin"2t dt"`. `I=inttsin"2t dt"=-(1)/(2)t cos2t+(1)/(4)sin2t+C` `=(1)/(2)t(2sin^(2)t-1)+(1)/(2)sintcost=(1)/(2)sin^(-1)sqrt(x)(2x-1)+(1)/(2)sqrt(x(1-x))+C`. |
|
| 58. |
`inttan^5xdx`A. `(1)/(6)tan^(6)x+C`B. `(1)/(4)tan^(4)x+(1)/(2)tan^(2)x+log|secx|+C`C. `(1)/(4)tan^(4)x-(1)/(2)x+log|secx|+C`D. none of these |
|
Answer» Correct Answer - C `I=inttan^(3)xtan^(2)xdx=inttan^(3)x(sec^(2)x-1)dx`. `=inttan^(3)xsec^(2)xdx-inttan^(3)xdx=intt^(3)dt-inttanx(sec^(2)x-1)dx`, where tan x=t `=(t^(4))/(4)-inttanxsec^(2)xdx+inttanxdx` `=(1)/(4)tan^(4)x-intudu+log|secx|+C`, where tan x=u `=(1)/(4)tan^(4)x-(1)/(2)u^(2)+log|secx|+C=(1)/(4)tan^(4)x-(1)/(2)tan^(2)x+log|secx|+C`. |
|
| 59. |
Evaluate : (i) `intxsec^(2)xdx` (ii) `intxsin2xdx` |
|
Answer» (i) Integrating by parts, taking x as the first function, we have `intxsec^(2)xdx=x*intsec^(2)xdx-int{(d)/(dx)(x)*intsec^(2)xdx}dx` `=xtanx-int1*tanxdxtanx+log|cosx|+C`. (ii) Integrating by parts, taking x as the first function, we get `intxsin2x=x*intsin2xdx-int{(d)/(dx)(x)*intsin2xdx}dx` `x*((-cos2x)/(2))-int1*((-cos2x)/(2))dx` `=(-xcos2x)/(2)+(1)/(2)intcos2xdx` `=(-xcos2x)/(2)+(1)/(2)*(sin2x)/(2)+C` `(-xcos2x)/(2)+(1)/(4)sin2x+C`. |
|
| 60. |
Evaluate : `intx^(2)sinxdx`. |
|
Answer» Integrating by parts, taking `x^(2)` as the first function, we get `intx^(2)sinxdx=x^(2)intsinxdx-int[(d)/(dx)(x^(2))*intsinxdx]dx` `=x^(2)(-cosx)-int2x(-cosx)dx` `=-x^(2)cosx+2intxcosxdx` `=-x^(2)cosx+2[x(sinx)-int{(d)/(dx)(x)*intcosxdx}dx]` [integrating x cos x by parts] `=-x^(2)cosx+2[xsinx-intsinxdx]` `=-x^(2)cosx+2[xsinx+cosx]+C`. |
|
| 61. |
Evaluate : `intx^(n)logxdx`. |
|
Answer» (i) Integrating by parts, taking x as the first function, we get `intx^(n)logx=(logx)*intx^(n)dx-int{(d)/(dx)(logx)*intx^(n)dx}dx` `=(logx)*(x^(n+1))/((n+1))-int(1)/(x)*(x^(n+1))/((n+1))dx` `=(x^(n+1)logx)/((n-1))-(1)/((n+1))intx^(n)dx` `=(x^(n+1)logx)/((n-1))-(x^(n+1))/((n+1)^(2))+C`. |
|
| 62. |
`int(e^(sqrt(x))cos(e^(sqrt(x))))/(sqrt(x))dx=?` |
|
Answer» Correct Answer - `2sin(e^(sqrt(x)))+C` Put `e^(sqrt(x))=t`. |
|
| 63. |
`int (x^2+1)/(x^4+1) dx` |
|
Answer» Correct Answer - `(1)/(sqrt(2))tan^(-1){(1)/(sqrt(2))(x-(1)/(x))}+C` Divide num. and denom. By `x^(2).Put(x-(1)/(x))=t`. `:.I=int(dt)/({t^(2)+(sqrt(2))^(2)})`. |
|
| 64. |
`int sqrt(e^x-1)dx` |
|
Answer» Correct Answer - `2sqrt(e^(x)-1)-2tan^(-1)sqrt(e^(x)-1)+C` Put `(e^(x)-1)=t^(2)anddx=(2t)/((t^(2)+1))` dt. Then, `I=int(2t^(2))/((1+t^(2)))dt=2int(1-(1)/(1+t^(2)))dt`. |
|
| 65. |
Evaluate:`int (dx)/(x^(1/2)+x^(1/3))` |
|
Answer» Correct Answer - `2sqrt(x)-3x^(1//3)+6x^(1//6)-6log|1+x^(1//6)|+C` `I=int(dx)/(x^(1//3)(1+x^(1//6)))*"Now Put "x=t^(6)anddx=6t^(5)dt`. Then, =`I=6int(t^(3))/(1+t)dt=6*int(t^(2)-t+1-(1)/(1+t))dt`. |
|
| 66. |
`int(dx)/((x-sqrt(x)))` |
|
Answer» Correct Answer - `2log|sqrt(x)-1|+C` `I=int(dx)/(sqrt(x)(sqrt(x)-1))*Putsqrt(x)-1=t and(1)/(sqrt(x))dx=2dt`. |
|
| 67. |
Evaluate the following integrals: `intxcosxdx` |
| Answer» Correct Answer - `xsinx+cosx+C` | |
| 68. |
Evaluate the following integrals: `intxsin3xdx` |
| Answer» Correct Answer - `(-xcos3x)/(3)+(sin3x)/(9)+C` | |
| 69. |
Evaluate the following integrals: `intxlog2xdx` |
| Answer» Correct Answer - `(1)/(2)x^(2)log2x-(1)/(4)x^(2)+C` | |
| 70. |
Evaluate:`int(x^2tan^(-1)x^3)/(1+x^6)dx`A. `(1)/(3)(tan^(-1)x^(3))^(2)+C`B. `log|tan^(-1)x^(3)|+C`C. `(1)/(6)(tan^(-1)x^(3))^(2)+C`D. none of these |
|
Answer» Correct Answer - C Put `tan^(-1)x^(3)=t and(3x^(2))/((1+x^(6)))dx=dt`. `:." "I=(1)/(3)intdt=(1)/(6)(tan^(-1)x^(3))^(2)+C`. |
|
| 71. |
`int(sec^(2)(2tan^(-1)x))/((1+x^(2)))dx` |
|
Answer» Correct Answer - `(1)/(2)tan(2tan^(-1)x)+C` Put `2tan^(-1)x=t`. |
|
| 72. |
Evaluate the following integrals: `intxe^(2x)dx` |
| Answer» Correct Answer - `(1)/(2)xe^(2x)-(1)/(4)e^(2x)+C` | |
| 73. |
Evaluate the following integrals: `intxe^(x)dx` |
| Answer» Correct Answer - `e^(x)(x-1)+C` | |
| 74. |
`intxe^(x)dx=?`A. `e^(x)(1-x)+C`B. `e^(x)(x+1)+C`C. `e^(x)(x-1)+C`D. none of these |
|
Answer» Correct Answer - C `intunderset(I)(x)underset(II)(e^(x))dx` |
|
| 75. |
`int (sin (2tan^(-1)x))/((1+x^(2)))dx.` |
| Answer» Correct Answer - `-(1)/(2)cos(2tan^(-1)x)+C` | |
| 76. |
Evaluate the following integrals: `intxcos2xdx` |
| Answer» Correct Answer - `(1)/(2)xsin2x+(1)/(4)cos2x+C` | |
| 77. |
`intxe^(2x)dx=?`A. `(1)/(2)xe^(2x)+(1)/(4)e^(2x)+C`B. `(1)/(2)xe^(2x)-(1)/(4)e^(2x)+C`C. `2xe^(2x)+4e^(2x)+C`D. none of these |
|
Answer» Correct Answer - B `intunderset(I)(x)underset(II)(e^(2x))dx` |
|
| 78. |
Evaluate : `int(cosx)/(("cos"(x)/(2)+"sin"(x)/(2))^(3))dx`. |
|
Answer» `int(cosx)/(("cos"(x)/(2)+"sin"(x)/(2))^(3))dx=int(cos^(2)(x//2)-sin^(2)(x//2))/({cos(x//2)+sin(x//2)}^(3))dx` `=int(cos(x//2)-sin(x//2))/(("cos"(x)/(2)+"sin"(x)/(2))^(2))dx=2int(1)/(t^(2))dt," where t=cos"(x)/(2)+"sin"(x)/(2)` `=(-2)/(t)+C=(-2)/(cos(x//2)+sin(x//2))+C` |
|
| 79. |
`intxsin2xdx=?`A. `(1)/(2)xcos2x+(1)/(4)sin2x+C`B. `-(1)/(2)xcos2x-(1)/(4)sin2x+C`C. `-(1)/(2)xcos2x+(1)/(4)sin2x+C`D. none of these |
|
Answer» Correct Answer - C `intunderset(I)(x)underset(II)(sin2x)dx` |
|
| 80. |
`intxcos2xdx=?`A. `(1)/(2)xsin2x+(1)/(4)cos2x+C`B. `(1)/(2)xsin2x-(1)/(4)cos2x+C`C. `2xsin24cos2x+C`D. none of these |
|
Answer» Correct Answer - A `intunderset(I)(x)underset(II)(cos2x)dx` |
|
| 81. |
`int(sin^(-1)x)/(sqrt(1-x^(2)))dx` |
| Answer» Correct Answer - `(1)/(2)(sin^(-1)x)^(2)+C` | |
| 82. |
`int(sqrt(cosx))sinxdx` |
| Answer» Correct Answer - `-(2)/(3)(cosx)^(3//2)+C` | |
| 83. |
`intxsec^(2)xdx=?`A. `xtanx-log|cosx|+C`B. `xtanx+log|cosx|C`C. `xtanx+log|secx|+C`D. none of these |
|
Answer» Correct Answer - B `intunderset(I)(x)underset(II)(sec^(2))xdx` |
|
| 84. |
Evaluate : `int(sinx)/(sqrt(1+sinx))dx`. |
|
Answer» `int(sinx)/(sqrt(1+sinx))dxint((1+sinx)-1)/(sqrt(1+sinx))dx` `intsqrt(1+sinx)dx-int(dx)/(sqrt(1+sinx))` `=int(dx)/sqrt(cos^(2)(x//2)+sin^(2)(x//2)+2sin(x//2)cos(x//2))` `=int[cos(x//2)+sin(x//2)]dx-int(dx)/([cos(x//2)+sin(x//2)])` `=(2"sin"(x)/(2)-2"cos"(x)/(2))-(1)/(sqrt(2))*int(dx)/((1)/(sqrt(2))"cos"(x)/(2)+(1)/(sqrt(2))"sin"(x)/(2))` `=(2"sin"(x)/(2)-2"cos"(x)/(2))-(1)/(sqrt(2))*int(dx)/(sin((x)/(2)+(pi)/(4)))` `=(2"sin"(x)/(2)-2"cos"(x)/(2))-(1)/(sqrt(2))int"cosec"((x)/(2)+(pi)/(4))dx` `=2("sin"(x)/(2)-"cos"(x)/(2))-(1)/(sqrt(2))xx2log|{:tan((x)/(4)+(pi)/(8)):}|+C` `=2("sin"(x)/(2)-"cos"(x)/(2))-sqrt(2)log|{:tan((x)/(4)+(pi)/(8)):}|+C`. |
|
| 85. |
Evaluate : `int(dx)/(sqrt(1-sinx))` |
|
Answer» `int(dx)/(sqrt(1-sinx))=(dx)/([sin^(2)(x//2)+cos^(2)(x//2)-2"sin"(x)/(2)"cos"(x)/(2)]^(1//2))` `=int(dx)/(("sin"(x)/(2)-"cos"(x)/(2)))=(1)/(sqrt(2))int(dx)/(((1)/(sqrt(2))*"sin"(x)/(2)-"cos"(x)/(2)*(1)/(sqrt(2))))` `=(1)/(sqrt(2))*int(dx)/(("sin"(x)/(2)"cos"(pi)/(4)-"cos"(x)/(2)"sin"(pi)/(4)))` `=(1)/(sqrt(2))int"cosec"((pi)/(2)-(pi)/(4))dx=(1)/(sqrt(2))2*log[tan((x)/(4)-(pi)/(8))]+C` `=sqrt(2)logtan((x)/(4)-(pi)/(8))+C`. |
|
| 86. |
Evaluate the following integrals:`intx^(3)cosx^(2)dx` |
|
Answer» Correct Answer - `(1)/(12)x^(2)sinx^(2)+(1)/(2)cosx^(2)+C` Put `x^(2)=t`. |
|
| 87. |
Evaluate the following integrals: `intxsinxcosxdx` |
|
Answer» Correct Answer - `-(1)/(4)xcos2x+(1)/(8)2x+C` `I=(1)/(2)intsin2xdx`. |
|
| 88. |
Evaluate the following integrals: `int(sin^(2)x)/((1+cosx)^(2))dx` |
|
Answer» Correct Answer - `2"tan"(x)/(2)-x+C` `I=int((1-cos^(2)x))/((1+cosx)^(2))dx=int((1-cosx))/((1+cosx))dx=int"tan"^(2)(x)/(2)dx=int(sec^(2)(x)/(2)-1)dx`. |
|
| 89. |
Evaluate the following integrals: `intcos^(3)xsin2xdx` |
|
Answer» Correct Answer - `-(2)/(5)cos^(5)x+C` `I=2intcos^(4)xsinxdx.Putcosx=t`. |
|
| 90. |
Evaluate the following integrals: `intsinxlog(cosx)dx` |
|
Answer» Correct Answer - `-cosxlog(cosx)+cosx+C` Put `cosx=t`. |
|
| 91. |
`intcossqrt(x)dx=?`A. `sinsqrt(x)+cossqrt(x)+C`B. `(1)/(2)(sqrt(x)sinsqrt(x)-cossqrt(x))+C`C. `2[sqrt(x)sinsqrt(x)+cossqrt(x)]+C`D. none of these |
|
Answer» Correct Answer - C Putting `sqrt(x)=t and(1)/(sqrt(x))dx=2dt", we get "dx=2tdt` `:." "I=2intunderset(I)(t)underset(II)(cost)dt`. |
|
| 92. |
Evaluate the following integrals: `int(cos^(9)x)/(sinx)dx` |
|
Answer» Correct Answer - `log|sinx|-2sin^(2)x+(3)/(2)sin^(4)x-(2)/(3)sin^(6)x+(1)/(8)sin^(8)x+C` `I=int(cos^(8)xcosx)/(sinx)dx=int((1-sin^(2)x)cosx)/(sinx)dx`. Put sin x=t. |
|
| 93. |
`inte^(sqrt(x))dx=?`A. `e^(sqrt(x))+sqrt(x)+C`B. `(1)/(2)e^(sqrt(x))(sqrt(x)+1)+C`C. `2e^(sqrt(x))(sqrt(x)-1)+C`D. none of these |
|
Answer» Correct Answer - C Putting `sqrt(x)=t and(1)/(2sqrt(x))dx=dt," we get "I=2intunderset(I)(t)underset(" "II)(" "e^(t))dt` |
|
| 94. |
`intsinxlog(cosx)dx=?` |
|
Answer» Correct Answer - `cosx(1-logcosx)+C` Put cos x=t. |
|
| 95. |
Evaluate the following integrals: `int(x)/((1+sinx))dx` |
|
Answer» Correct Answer - `-xtan((pi)/(4)-(x)/(2))+2log|cos((pi)/(4)-(x)/(2))|+C` `I=int(x)/(1+cos((pi)/(2)-x))dx=(1)/(2)intxsec^(2)((pi)/(4)-(x)/(2))dx` |
|
| 96. |
`int(logx)^(2)dx=?`A. `(2logx)/(x)+C`B. `(1)/(3)(logx)^(3)+C`C. `x(logx)^(2)-2xlogx+2x+C`D. `x(logx)^(2)-2xlogx-2x+C` |
|
Answer» Correct Answer - C `I={underset(I)((logx)^(2))*underset(II)(1)}dx=(logx)^(2)*x-int(2logx)/(x)*xdx` `=x(logx)^(2)-2intlogxdx=x(logx)^(2)-2[x(logx-1)]+C`. |
|
| 97. |
Evaluate the following integrals: `intlog(2+x^(2))dx` |
|
Answer» Correct Answer - `xlog(x^(2)+2)-22x+2sqrt(2)tan^(-1)((x)/(sqrt(2)))+C` Take `log(2+x^(2))" as 1st function". |
|
| 98. |
Evaluate : `int(logx)^(2)dx`. |
|
Answer» Integrating by parts, taking `(logx)^(2)` as the first function and 1 as the second function, we get `int(logx)^(2)dx=int{(logx)^(2)*1}dx` `=(logx)^(2)*intdx-int{(d)/(dx)(logx)^(2)*int1dx}dx` ]`=x(logx)^(2)-int((2logx)/(x)*x)dx` `=x(logx)^(2)-2int(logx*1)dx` `=x(logx)^(2)-2[(logx)intdx-int{(d)/(dx)(logx)*intdx}dx]` `=x(logx)^(2)-2[xlogx-int(1)/(x)*xdx]` `=x(logx)^(2)-2xlogx+2x+C`. |
|
| 99. |
`intlog (1+x^(2))dx.` |
|
Answer» Integrating by parts, taking log `(1+x^(2))` as the first function and 1 as the second function, we get `intlog(1+x^(2))dx=int{log(1+x^(2))*1}dx` `=log(1+x^(2))*intdx-int[(d)/(dx){log(1+x^(2))}*int1dx]dx` `=log(1+x^(2))*x-int(2x)/((1+x^(2)))*xdx` `=xlog(1+x^(2))-2int(x^(2))/((1+x^(2)))dx` `=log(1+x^(2))-2int(1-(1)/(1+x^(2)))dx` `=xlog(1+x^(2))-2intdx+2int(dx)/((1+x^(2)))` `=xlog(1+x^(2))-2x+2tan^(-1)x+C`. |
|
| 100. |
`int(logx)/(x^(2))dx=?`A. `-(1)/(x)(logx+1)+C`B. `(1)/(x)(logx-1)+C`C. `(1)/(x)(logx+1)+C`D. none of these |
|
Answer» Correct Answer - A `int(logx)/(x^(2))dx=intunderset(I)((logx))*underset(II)((1)/(x^(2)))dx` |
|