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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.
| 51. |
The value of ` int(1+logx)/(sqrt((x^(x))^(2)-1))dx " is " `A. `sec^(-1)(x^(x))+C`B. `log|x^(x)+sqrt(x^(2x)-1)|+C`C. `log|x^(x)-sqrt(x^(2x)-1)|+C`D. none of these |
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Answer» Correct Answer - a We have , `I=int(1+logx)/(sqrt(x^(2x)-1))dx` `=int(1)/(x^(x)sqrt((x^(x))^(2)-1^(2)))d(x^(x))=sec^(-1)(x^(x))+C` |
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| 52. |
`int((x4-x)^(1//4))/(x^(5))dx` is equal toA. `(4)/(15)(1-(1)/(x^(3)))^(5//4)+C`B. `(4)/(5)(1-(1)/(x^(3)))^(5//4)+C`C. `(4)/(15)(1+(1)/(x^(3)))^(5//4)+C`D. none of these |
| Answer» Correct Answer - a | |
| 53. |
Integrate the functions`fprime(a x+b)[f(a x+b)]^n`A. `(1)/(n+1){f(ax+b)}^(n+1)+C`, for all n except n =-1B. `(1)/(n+1){f(ax+b)}^(n+1)+C` , for all nC. `(1)/(a(n+1)}{f(ax+b)}^(n+1)+C` for all n except n =- 1D. `(1)/(a(n+1)){f(ax+b)}^(n+1)+C` , for all n |
| Answer» Correct Answer - c | |
| 54. |
If `int(xe^x)/sqrt(1+e^x)dx=f(x)sqrt(1+e^x)-2logg(x)+c`, thenA. `f(x)=x-1`B. `g(x)=(sqrt(1+e^(x))-1)/(sqrt(1+e^(x))-1)`C. `g(x)=(sqrt(1+e^(x))+1)/(sqrt(1+e^(x))-1)`D. `f(x)=2(x+2)` |
| Answer» Correct Answer - d | |
| 55. |
`int(1+x)/(1+3sqrt(x))dx` is equal toA. `(3)/(5)x^(5//3)-(3)/(4)x^(4//3)+x+C`B. `(3)/(5)x^(5//3)-(3)/(4)x^(4//3)+C`C. `(3)/(5)x^(5//3)-(3)/(4)x^(4//3)+C`D. none of these |
| Answer» Correct Answer - a | |
| 56. |
`int{log(logx)+(1)/((logx)^(2))}dx=x {f (x)-g(x)}+C`, thenA. `f(x)=log(logx),g (x)=(1)/(logx)`B. `f(x)=logx,g(x)=(1)/(logx)`C. `f(x)=(1)/(logx),(x)=log(logx)`D. `f(x)=(1)/(xlogx),g(x)=(1)/(logx)` |
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Answer» Correct Answer - a We have , `I=int{log(logx)+(1)/((logx)^(2))}dx` `rArr I= inte^(t)(logt+(1)/(t^(2)))dt`,where t = log x `rArrI=inte^(t)(logt+(1)/(t))dt+inte^(t)(-(1)/(t)+(1)/(t^(2)))dt` `rArrI=e^(t)logt +e^(t)(-(1)/(t))+C` `rArrI=x(log(logx)-(1)/(logx))+C` `:. f(x) =logx and g(x) = (1)/(logx)` |
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| 57. |
`int(sin^(3)x)/((1+cos^(2)x)sqrt(1+cos^(2)x+cos^(2)x+cos^(4))x)dx` is equal toA. `sec^(-1)(secx+cosx)+C`B. `sec^(-1)(secx-cosx)+C`C. `sec^(-1)(secx-tanx)+C`D. none of these |
| Answer» Correct Answer - a | |
| 58. |
If `int(2^(1//x))/(x^(2))dx=a2^(1//x)+C`, then a=A. `-log_(2)e`B. `-log_(e)2`C. `-1`D. `1//2` |
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Answer» Correct Answer - a `int(2^(1//x))/(x^(2))dx=-int2^(1//x)d((1)/(x))=(-2^(1//x))/(log_(e)2)+C=-(log_(2)e)2^(1//x)+C` `:. a=-log_(2)e` |
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| 59. |
The value of `int((x-x^(3))^(1//3))/(x^(4))dx` isA. `(3)/(8)((1)/(x^(2))-1)^(4//3)+C`B. `-(3)/(8)((1)/(x^(2))-1)^(4//3)+C`C. `(1)/(8)(1-(1)/(x^(2)))^(4//3)+1`D. none of these |
| Answer» Correct Answer - b | |
| 60. |
If `int(sqrt(5+x^(10)))/(x^(16))dx=a(1+(5)/(x^(10)))^(3//2)+C`,then a=A. `-(1)/(25)`B. `(1)/(75)`C. `-(1)/(75)`D. `-(1)/(150)` |
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Answer» Correct Answer - c We have , `I=int(sqrt(5+x^(10)))/(x^(16))dx` `rArrI=intsqrt((5+x^(10))/(x^(10)))xx(1)/(x^(11))dx` `rArr I=(-1)/(50)intsqrt(1+(5)/(x^(10)))xx(-50)/(x^(11))dx` `rArrI=-(1)/(50)intsqrt(1+(5)/(x^(10)))d(1+(5)/(x^(10)))` `rArrI=-(1)/(50)xx(2)/(3)(1+(5)/(x^(10)))^(3//2)+C=-(1)/(75)(1+(5)/(x^(10)))^(3//2)+C` `:. a=-(1)/(75)` |
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| 61. |
`int(dx)/(sqrt(sin^(3)xcosx))=?`A. `(-2)/(sqrt(tanx))+C`B. `2sqrt(tanx)+C`C. `(2)/(sqrt(tanx))+C`D. `-2sqrt(tanx)+C` |
| Answer» Correct Answer - a | |
| 62. |
If `int(1)/((sinx+4)(sinx-1))dx` `=A(1)/("tan"(x)/(2)-1)+B"tan"^(-1){f(x)}+C`. Then,A. `A=(1)/(5),B=(-2)/(5sqrt(15)),f(x)=(4tanx+3)/(sqrt(15))`B. `A=-(1)/(5),B=(1)/(sqrt(15)),f(x)=(4tan(x//2)+1)/(sqrt(15))`C. `A=(2)/(5),B=(-2)/(5),f(x)=(4tanx+1)/(5)`D. `A=(2)/(5),B=(-2)/(5sqrt(15)),f(x)=(4tanx//2+1)/(sqrt(15))` |
| Answer» Correct Answer - d | |
| 63. |
`int (x)^(1/3) (7sqrt(1+3sqrt(x^(4))))dx` is equal toA. `(21)/(32){1+root(3)(x^(4))}^(8//7)+C`B. `(32)/(21){1+root(3)(x^(4))}^(8//7)+C`C. `(7)/(32){1+root(3)(x^(4))}^(8//7)+C`D. none of these |
| Answer» Correct Answer - a | |
| 64. |
`int(1)/(x(x^(4)-1))dx` is equal toA. `(1)/(4)log|(x^(4))/(x^(4)-1)|+C`B. `(1)/(4)log|(x^(4)-1)/(x^(4))|+C`C. `log|(x^(4)-1)/(x^(4))|+C`D. `log|(x^(4))/(x^(4)-1)|+C` |
| Answer» Correct Answer - b | |
| 65. |
`int(x^(2))/((a+bx^(2))^(5//2))dx` is equal toA. `-(1)/(3a)((x^(2))/(a+bx^(2)))^(3//2)+C`B. `(1)/(3a)((x^(2))/(a+bx^(2)))^(3//2)+C`C. `(1)/(2a)((x^(2))/(a+bx^(2)))^(2//3)+C`D. none of these |
| Answer» Correct Answer - b | |
| 66. |
`Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C ,`then`f(x)=1/2x^2`(b) `g(x)=logx``A=1`(d) none of theseA. `f(x)=(1)/(2)x^(2)`B. `g(x)=logx`C. `A=1`D. none of these |
| Answer» Correct Answer - d | |
| 67. |
`int(2)/((e^(x)+e^(-x))^(2))dx` is equal toA. `(-e^(-x))/(e^(x)+e^(-x))+C`B. `-(1)/(e^(x)+e^(-x))+C`C. `-(1)/((e^(x)+1)^(2))+C`D. `(1)/(e^(x)-e^(-x))+C` |
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Answer» Correct Answer - a We have `I= int(2)/((e^(x)+e^(-x))^(2))dx` `=int (2e^(2x))/((e^(2x)+1)^(2))dx= int(1)/((e^(2x)+1)^(2))d(e^(2x)+1)` `rArr I=-(1)/(e^(2x)+1)+C=(-e^(-x))/(e^(x)+e^(-x))+C` |
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| 68. |
` int(x+2)/((x^(2)+3x+3)sqrt(x+1))dx " is equal to"`A. `(1)/(sqrt(3))tan^(-1)((x)/(sqrt(3(x+1))))`B. `(2)/(sqrt(3))tan^(-1)((x)/((sqrt(x+1))))`C. `(2)/(sqrt(3))tan^(-1)((x)/(sqrt(x+1)))`D. none of these |
| Answer» Correct Answer - b | |
| 69. |
The value of `int(1)/(x^(2)(x^(4)+1)^(3//4))dx` , isA. `(1+(1)/(x^(4)))^(1//4)`B. `-(1+(1)/(x^(4)))^(1//4)`C. `-(1)/(4)(1+(1)/(x^(4)))^(1//4)`D. none of these |
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Answer» Correct Answer - b We have , `I=int(1)/(x^(2)(x^(4)+1)^(3//4))dx` `rArr I=int(1)/(x^(5)(1+(1)/(x^(4)))^(3//4))dx` `rArr I=-(1)/(4)int(1+(1)/(x^(4)))^(-3//4)((-4)/(x^(5)))dx` `rArr I=-(1)/(4)int(1+(1)/x^(4))^(-3//4)d(1+(1)/(x^(4)))` `rArrI=-{(1+(1)/(x^(4)))^(1//4)/(1//4)}+C=-(1+(1)/(x^(4)))^(1//4)+C` |
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| 70. |
Let `f(x)=int(1)/((1+x^(2))^(3//2))dx` and f(0)=0 then f(1)=A. `-(1)/(sqrt(2))`B. `(1)/(sqrt(2))`C. `sqrt(2)`D. none of these |
| Answer» Correct Answer - b | |
| 71. |
`int(1)/((a^(2)+x^(2))^(3//2))dx` is equal toA. `(x)/(a^(2)sqrt(a^(2)+x^(2)))+C`B. `(x)/((a^(2)+x^(2))^(3//3))+C`C. `(1)/(a^(2)sqrt(a^(2)+x^(2)))+C`D. none of these |
| Answer» Correct Answer - a | |
| 72. |
`int(1)/(x^(1//2)(1+x^(2))^(5//4))dx` is equal toA. `(-2sqrt(x))/(4sqrt(1+x^(2)))+C`B. `(2sqrt(x))/(4sqrt(1+x^(2)))+C`C. `(-sqrt(x))/(4sqrt(1+x^(2)))+C`D. `(sqrt(x))/(4sqrt(1+x^(2)))+C` |
| Answer» Correct Answer - b | |
| 73. |
If `intf(x)sinxcosxdx=(1)/(2(b^(2)-a^(2)))log{f(x)}+C` then f(x) is equal toA. `(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)`B. `(1)/(a^(2)sin^(2)x-b^(2)cos^(2)x)`C. `(1)/(a^(2)cos^(2)x+b^(2)sin^(2)x)`D. `(1)/(a^(2)cos^(2)x-b^(2)sin^(2)x)` |
| Answer» Correct Answer - a | |
| 74. |
The primitive of the function `f(x)=(1-(1)/(x^(2)))a^(x+(1)/(x))x,gt0`, isA. `(a^(x^+(1)/(x)))/(log_(e)a)`B. `a^(x+(1)/(x))log_(e)a`C. `a^(x+(1)/(x))/(x)log_(e)a`D. `a^(x+(1)/(x))/(log_(e)a)` |
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Answer» Correct Answer - a The primitive of f (x) is `int(1-(1)/(x^(2)))a^(x+(1)/(x))dx = int (x+(1)/(x))=(a^(x+(1)/(x)))/(log_(e)a)+C` |
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| 75. |
The primitive of the function f (x) `=(2x+1)|sinx|`, where `piltxlt2pi` isA. `-(2x+1)cosx+2sinx+C`B. `(2x+1)cosx-2sin x+C`C. `(x^(2)+x)cosx+C`D. none of these |
| Answer» Correct Answer - b | |
| 76. |
`int(dx)/(x(x^n+1))` is equal toA. `(1)/(n)log((x^(n))/(x^(n)+1))+C`B. `(1)/(n)log((x^(n)+1)/(x^(n)))`C. `log((x^(n))/(x^(n)+1))+C`D. none of these |
| Answer» Correct Answer - a | |
| 77. |
`int3sqrt((sin^n x)/(cos^(n+6) x))dx`A. `(3)/(n+3)tan^(n//3+1)x+C`B. `(3)/(n+3)tan^(n//3+1)x+C`C. `(3)/(n+1)tan^(n//3+1)x+C`D. none of these |
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Answer» Correct Answer - b `I=int3sqrt((sin^(n)x)/(cos^(n+6)x))dx=int3sqrt(tan^(n))xsec^(2)xdx` `rArr I= inttan^(n//3)xd ((tanx)^(n/(3)+1))/(n/(3)+1)=(3)/(n+3)tan^(n/(3)+1)x+C` |
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| 78. |
`int(x-1)/((x+1)sqrt(x^(3)+x^(2)+x))dx` is equal toA. `tan^(-1)sqrt((x^(2)+x+1)/(x))+C`B. `2tan^(-1)sqrt((x^(2)+x+1)/(x))+C`C. `3tan^(-1)sqrt((x^(2)+x+1)/(x))+C`D. none of these |
| Answer» Correct Answer - b | |
| 79. |
`int(1+x^(4))/((1-x^(4))^(3//2))dx` is equal toA. `(x)/(sqrt(1-x^(4)))+C`B. `(-x)/(sqrt(1-x^(4)))+C`C. `(2x)/(sqrt(1-x^(4)))+C`D. `(-2x)/(sqrt(1-x^(4)))+C` |
| Answer» Correct Answer - a | |
| 80. |
`int(1)/(sqrt(x^(2)+2))d(x^(2)+1)` is equal toA. `2sqrt(x^(2)+2)+C`B. `2sqrt(x^(2)+2)+C`C. `(1)/((x^(2)+2)^(3//2))+C`D. none of these |
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Answer» Correct Answer - a We have , `I=int(1)/(sqrt(x^(2)+2))d(x^(2)+1)` `rArrI=int(1)/(sqrt(x^(2)+2))d(x^(2)+2)` " " `[becaused(x^(2)+1)=d(x^(2)+2)]` `rArrI=2sqrt(x^(2)+2)+C` |
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| 81. |
The primitive of the function f (x) `=(2x+1)|cosx|`, when `(pi)/(2)ltxltpi` is given byA. `cosx+x sin x`B. `-cosx-xsinx`C. `xsinx-cos x`D. none of these |
| Answer» Correct Answer - b | |
| 82. |
Given f(x) `=|{:(0,x^(2)-sinx,cosx-2),(sinx-x^(2),0,1-2x),(2-cosx,2x-1,0):}|` `intf(x)` dx is equal toA. `(x^(3))/(3)-x^(2)sinx+sin2x+C`B. `(x^(3))/(3)-x^(2)sin x-cos 2x+C`C. `(x^(3))/(3)-x^(2)cosx-cos2x+C`D. none of these |
| Answer» Correct Answer - d | |
| 83. |
The value of `int(x+(1)/(x))^(3//2)((x^(2)-1)/(x^(2)))dx` , isA. `(2)/(3)(x+(1)/(x))^(3//2)+C`B. `(2)/(5)(x+(1)/(x))^(5//2)+C`C. `2(x+(1)/(x))^(1//2)+C`D. none of these |
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Answer» Correct Answer - b We have , `I =int(x+(1)/(x))^(3//2)((x^(2)-1)/(x^(2))dx` `rArrI=int(x+(1)/(x))^(3//2)d(x+(1)/(x))=(2)/(5)(x+(1)/(x))^(5//2)+C` |
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| 84. |
`int (cos2x)/(cosx) dx=`A. `2sinx+log(secx-tanx)+C`B. `2sinx-log(secx-tanx)+C`C. `2sinx+log(secx+tanx)+C`D. none of these |
| Answer» Correct Answer - a | |
| 85. |
`int(1+x^(2))/(xsqrt(1+x^(4)))dx` is equal toA. `-log|x-(1)/(x)+sqrt((x-(1)/(x))^(2))-2|+C`B. `-log|x-(1)/(x)+sqrt((x-(1)/(x))^(2))+2|+C`C. `-log|x-(1)/(x)+sqrt((x-(1)/(x))^(2))-2|+C`D. none of these |
| Answer» Correct Answer - b | |
| 86. |
The value of `int((ax^2-b)dx)/(xsqrt(c^2x^2-(ax^2+b)^2))` is equal toA. `sin^(-1)((ax+(b)/(x))/(c))+k`B. `sin^(-1)((ax^(2)+(b)/(x^(2)))/(c))+k`C. `cos^(-1)((ax+b//x)/(c))+k`D. `cos^(-1)((ax^(2)+(b)/(x^(2)))/(c))+k` |
| Answer» Correct Answer - a | |
| 87. |
`int(x^(2-1))/(xsqrt(x^4+3x^2)+1)`dx is equal toA. `log_(e)|x+(1)/(x)+sqrt(x^(2)+(1)/(x^(2))+3)|+C`B. `log_(e)|x-(1)/(x)+sqrt(x^(2)+(1)/(x^(2))-3)|+C`C. `log_(e)|x+sqrt(x^(2)+3)|+C`D. none of these |
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Answer» Correct Answer - a We have , `I=int(x^(2)-1)/(sqrt(x^(4)+3x^(2)+1))dx=int(1-(1)/(x^(2)))/(sqrt(x^(2)+3+(1)/(x^(2))))dx` `rArrI=int(1)/(sqrt((x+(1)/(x))^(2)+1^(2)))d(x+(1)/(x))` `rArrI=log_(e)|x+(1)/(x)+sqrt((x+(1)/(x))^(2)+1)|+C` |
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| 88. |
Integration of f (x) `= sqrt(1+x^(2))` with respect to `x^(2)`, isA. `(2)/(3)((1+x^(2))^(3//2))/(x)+C`B. `(2)/(3)(1+x^(2))^(3//2)+C`C. `(2x)/(3)(1+x^(2))^(3//2)+C`D. none of these |
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Answer» Correct Answer - b Let `I=intf(x)d (x^(2))=intsqrt(1+x^(2))d(x^(2))` `rArrI=intsqrt(1+x^(2))d(1+x^(2))` " " `[becaused(x^(2))=d(1+x^(2))]` `rArrI=(2)/(3)(1+x^(2))^(3//2)+C` |
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| 89. |
`intsinx d (cosx)` is equal toA. `(1)/(2)sin2x-x+C`B. `(1)/(2)((1)/(2)sin2x-x)+C`C. `(1)/(2)((sin2x)/(2)+x)+C`D. none of these |
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Answer» Correct Answer - b `d(cosx) =(d)/(dx)(cosx)*dx =-sinx dx` `:. intsin xd (cosx)=-intsin^(2)x dx =-(1)/(2)(1-cos2x)dx` `rArrintsinx d (cos x) =-(1)/(2)x-((sin2x)/(2))+C` `=(1)/(2)((1)/(2)sin2x-x)+C` |
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| 90. |
`int(1)/(xsqrt(1-x^(3)))`dx is equal toA. `(1)/(3)log|(sqrt(1-x^(3)-1))/(sqrt(1-x^(3))+1)|+C`B. `(1)/(2)log|(sqrt(1-x^(2))+1)/(sqrt1-x^(2))-1|+C`C. `(1)/(3)log|(1)/(sqrt(1-x^(3)))|+C`D. none of these |
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Answer» Correct Answer - a We have , `I=int(1)/(xsqrt(1-x^(3)))dx` `rArrI=-(1)/(3)int(1)/(x^(3)sqrt(1-x^(3)))(-3x^(2))dx` `rArrI=-(1)/(3)int(1)/(x^(3)sqrt(1-x^(3)))(1-x^(3))dx` `rArrI=-(1)/(3)int(1)/((1-t^(2))sqrt(t^(2)))` 2t dt , where `t^(2)=1-x^(3)` `rArrI=-(2)/(3)int(1)/(1-t^(2))dt=(2)/(3)int(1)/(t^(2)-1^(2))dt=(1)/(3)log|(t-1)/(t+1)|+C` `rArrI=(1)/(3)log|(sqrt(1-x^(3))-1)/(sqrt(1-x^(3))+1)|+C` |
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| 91. |
`int(e^((x^(2)+4Inx))-x^(3)e^(x^(2)))/(x-1)dx` equals toA. `((e^(3 Inx)-e^(Inx))/(2x))e^(x^(2))+C`B. `((x-1)xe^(x^(2)))/(2)+C`C. `((x^(2)-1))/(2x)e^(x^(2))+C`D. none of these |
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Answer» Correct Answer - d Let `I=int(e^(x^(x)+4Inx)-x^(3)e^(x^(2)))/(x-1)dx=int(e^(x^(2)).x^(4)-x^(3)e^(x^(2)))/(x-1)dx` `rArrI=intx^(3)e^(x^(2))dx=(1)/(2)intte^(t)dt`, where `t=x^(2)` `I=(1)/(2)(t-1)e^(t)+C=(1)/(2)(x^(2)-1)e^(x^(2))+C` |
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| 92. |
If `intx log (1+(1)/(x))dx` `=f(x).log_(e)(x+1)+g(x)log_(e)x^(2)xLx+C` , thenA. `f(x)=(x^(2))/(2)`B. `g(x)=log_(e)x`C. L = 1D. `L=(1)/(2)` |
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Answer» Correct Answer - d Let `I=intx log(1+(1)/(x))dx` `rArrI=intunderset(II)(x)logunderset(I" ")((1+x))dx-intunderset(II)(x)logx dx` `rArr I=(x^(2))/(2)log(1+x)-(1)/(2)int(x^(2))/(x+1)dx-{(x^(2))/(2)logx-int(1)/(x)xx(x^(2))/(2)dx}` `rArr I=(x^(2))/(2)log_(e)(1+x)-(1)/(2)intx-1+(1)/(x+1)dx-(x^(2))/(2)log_(e)x+(x^(2))/(4)+C` `rArrI=(x^(2))/(2)log_(e)(1+x)-(1)/(2)((x^(2))/(2)-x)-(1)/(2)log_(e)(x-1)-(x^(2))/(2)log_(e)x+(x^(2))/(4)+C` `rArrI=((x^(2)-1)/(2))log_(e)(1+x)-(x^(2))/(2)log_(e)x+(x)/(2)+C` Hence , f(x)`=(x^(2)-1)/(2) ,g(x)=-(x^(2))/(2)andL=(1)/(2)` |
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| 93. |
`intx^(x)(1+log_(e)x)`dx is equal toA. `x^(x)log_(e)x+C`B. `ex^(x)+C`C. `x^(x)+C`D. none of these |
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Answer» Correct Answer - c We have , `intx^(x)(1+log_(e)x)dx=int1.d(x^(x))=x^(x)+C` |
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| 94. |
If `f((3x-4)/(3x+4))=x+2`, then `int` f(x)dx is equal toA. `e^(x+2)log_(e)|(3x-4)/(3x+4)|`B. `-(8)/(3)log_(e)|1-x|+(2)/(3)x+C`C. `(8)/(3)log_(e)|x-1|+(x)/(3)+C`D. none of these |
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Answer» Correct Answer - b We have , `f((3x-4)/(3x+4))=x+2` Let `(3x-4)/(3x+4)=alpha` ` rArr ((3x-4)+(3x+4))/((3x-4)-(3x+4))=(alpha+1)/(alpha-1)` `rArr(6x)/(-8)=(alpha+1)/(alpha-1)` `rArrx=-(4)/(3)((alpha+1)/(alpha-1))` `rArrx+2=-(4alpha+4)/(3alpha-3)+2=(-4alpha-4+6alpha-6)/(3alpha-3)=(2alpha-10)/(3alpha-3)` `becausef((3x-4)/(3a+4))=x+2` `rArrf(alpha)=(2alpha-10)/(3alpha-3)` `rArrf(alpha)=(2)/(3)((alpha-5)/(alpha-1))` `rArrf(alpha)=(2)/(3)((alpha-1-4)/(alpha-1))=(2)/(3)(1-(4)/(alpha-1))=(2)/(3)-(8)/(3(alpha-1))` `rArrf(x)=(2)/(3)-(8)/(3(x-1))` `rArr intf(x)dx=int{(2)/(3)-(8)/(3(x-1))}dx` `becauseintf(x)dx=(2)/(3)x-(8)/(3)log_(e)|x-1|+C` |
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| 95. |
If `int(sqrt(cotx))/(sinxcosx)dx=Psqrt(cotx)+Q` , then P equalsA. 1B. 2C. `-1`D. `-2` |
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Answer» Correct Answer - d Let `I=int(sqrt(cotx))/(sinxcosx)dx` `rArrI=int(sqrt(cotx))/(tanx)sec^(2)xdx` " " `["Dividing"N^(r)and D^(r)"by "cos^(2)x]` `rArrI=int(tanx)^(-3//2)d(tanx)` `rArrI=-2(tanx)^(-1//2)+Q=-2sqrt(cot)x+Q` Hence , P =-2 |
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| 96. |
Evaluate the following integrals :`int(5cos^3x+6sin^3x)/(2sin^2xcos^2x)dx` |
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Answer» `I = int (5cos^3x+6sin^3x)/(2sin^2xcos^2x) dx` `=> I = int (5/2 cosx/sin^2x + 3sinx/cos^2x) dx` `=> I = int (5/2 cotxcosecx + 3 tanxsecx) dx` `=>I = 5/2(-cosecx) +3(secx) +c` `=>I =-5/2cosecx +3secx +c` |
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| 97. |
If n is a positive odd integer, then `int |x^n| dx=`A. `|(x^(n+1))/(n+1)|+C`B. `(x^(n+1))/(n+1)+C`C. `(|x^(n)|)/(n+1)+C`D. none of these |
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Answer» Correct Answer - c We have the following cases : CASE I When `xge0` In this case , we have , `int|x^(n)|dx=int|x|^(n)dx` `rArrint|x^(n)|dx=intx^(n)dx " " [because|x|=x]` `rArrint|x^(n)|dx=(x^(n+1))/(n+1)+C` `rArrint|x^(n)|dx=(|x|^(n)x)/(n+1)+C " " [becausexge0therefore|x|=x]` CASE II When ` xle0` In this case , we have `|x| =-x` `thereforeint|x^(n)|dx= int|x|^(n)dx=int (-x)^(n)dx` `rArrint|x^(n)|dx=-intx^(n)dx " " [because n is odd"]` `rArrint|x^(n)|dx=-(x^(n)+1)/(n+1)+C` `rArr int|x^(n)|dx=((-x)^(n)x)/(n+1)+C " " [{:(because " n is odd",),(therefore(-x)^(n)=-x^(n),):}]` `rArrint|x^(n)|dx=(|x|^(n)x)/(n+1)+C` Hence , `int|x^(n)|dx=(|x|^(n)x)/(n+1)+C` |
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| 98. |
If `inte^(ax)cosbx dx=(e2x)/(29)f(x)+C` , then f' (x)=A. 29 f (x)B. `-29 f(x)`C. 25 f(x)D. `-25 f(x)` |
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Answer» Correct Answer - d We have , `inte^(ax)cosbx dx=(e^(ax))/(a^(2)+b^(2))(a cos bx +b sinbx)+C` `rArr(e^(2x))/(2^(2)+5^(2))f(x)+C (e^(ax))/(a^(2)+b^(2))(a cos bx +b sin bx)+C` `rArr a=2 , b =5 and f(x) =a cos bx +b sin dx` `rArr f ' (x) =-b^(2)f(x)rArrf'(x)=-25f(x)` |
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| 99. |
But for all arbitrary constants, `intsqrt((1+sintheta-sin^(2)theta-sin^(3)theta)/(2sintheta-1))d theta` is equal toA. `(1)/(2)sqrt(sin theta-cos2 theta)+(3)/(4sqrt(2))log_(e)|(4sintheta+1)+2sqrt(2)sqrt(sintheta-cos2theta)|`B. `(1)/(2)sqrt(sintheta+cos2theta)+(3)/(4sqrt(2))log_(e)|(4sin theta -1)+2sqrt(2)sqrt(sintheta+cos2 theta)|`C. `(1)/(2sqrt(2))sqrt(sintheta-cos2 theta)+(3)/(4)log_(e)|(4 sin theta+1)-sqrt(sin theta - cos 2 theta)|`D. `(1)/(2)sqrt(sin theta+cos2theta)+(3)/(4sqrt(2))log_(e)|4sintheta+1-sqrt(sintheta-cos2 theta)|` |
| Answer» Correct Answer - a | |
| 100. |
Statement -1 : If `I_(1)=int(e^(x))/(e^(4x)+e^(2x)+1)dx` and `I_(2)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx`, then `I_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C` where C is an arbitrary constant. Statement -2 : A primitive of f(x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is `(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))`.A. Statement - 1 True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.B. Statement - 1 is True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.C. Statement - 1 True ,Statement - 2 is False.D. Statement - 1 is False , Statement - 2 is True. |
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Answer» A primitive of f (x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is given by `I=int(x^(2)=1)/(x^(4)+x^(2)+1)dx=int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2))+1)dx` `rArrI=int(1)/((x+(1)/(x))^(2)-1^(2))d(x+(1)/(x))=(1)/(2)log|(x+(1)/(x)-1)/(x+(1)/(x)+1)|+C` `rArrI=int(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))+C` So , statement - 2 is true. Now , `I_(2)-I_(1)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx` `rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx` `rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)d(e^(x))` `rArrI_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C` [ Using statment -2 ] So , statement - 1 is true . Also . statement - 2 is a correct explanation for statement -1. |
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