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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.
| 151. |
Evaluate : `inte^(2x)*(-sinx+2cosx)dx`. |
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Answer» We have `I=inte^(2x)*{cosx-sinx}dx=2inte^(2x)cosxdx-inte^(2x)sinxdx` `=2*{:[cosx*(e^(2x))/(2)-int(-sinx)*(e^(2x))/(2)dx]:}-inte^(2x)sinxdx` [integrating `e^(2x)` cos x by parts] `=e^(2x)cosx+inte^(2x)sinxdx-inte^(2x)sinxdx+C` `=e^(2x)cosx+C`. |
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| 152. |
Evaluate: `int(sin^(-1)x)^2 dx` |
| Answer» Correct Answer - `x(sin^(-1)x)^(2)+2sqrt(1-x^(2))sin^(-1)x-2x+C` | |
| 153. |
Evaluate the following integrals: `inte^(x)((2-sin2x)/(1-cos2x))dx` |
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Answer» Correct Answer - `-e^(x)cotx+C` `I=inte^(x){(2)/(2sin^(2)x)-(2sinxcosx)/(2sin^(2)x)}dx=inte^(x)(-cotx+"cosec"^(2)x)dx=-e^(x)cotx+C`. |
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| 154. |
Evaluate: `inte^x ((sin4x-4)/(1-cos4x)) dx` |
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Answer» Correct Answer - `e^(x)cot2x+C` `I=inte^(x){(2sin2xcos2x)/(2sin^(2)2x)-(4)/(2sin^(2)2x)}dx=inte^(x){cot2x-2"cosec"^(2)2x}dx=e^(x)cot2x+C`. |
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| 155. |
Evaluate the following integrals: `intcos^(4)xdx` |
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Answer» Correct Answer - `(3x)/(8)+(sin4x)/(32)+(sin2x)/(4)+C` `I=(1)/(4)int(2cos^(2)x)^(2)dx=(1)/(4)(1+cos2x)^(2)dx` `=(1)/(4)int{:[(1+2cos2x)+((1+cos4x)/(2))]:}dx=int{:[(3)/(8)+(1)/(2)cos2x+(1)/(8)cos4x]:}dx`. |
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| 156. |
Evaluate : (i) `intcos^(-1)xdx` (ii) `inttan^(-1)xdx` (iii) `intsec^(-1)xdx` |
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Answer» (i) Put `cos^(-1)x=t" so that "x=cost anddx=-sintdt`. `:.intcos^(-1)xdx=-inttsintdt` `=-[t*(-cost)-int*(-cost)dt]` [integrating by parts] `=tcost-intcostdt=tcost-sint+C` `=xcos^(-1)x-sqrt(1-x^(2))+C` `[becausecost=xrArrsint=sqrt(1-x^(2))]`. (ii) Put `tan^(-1)x=t" so that "x=tant anddx=sec^(2)tdt`. `:.inttan^(-1)xdx=inttsec^(2)tdt` `=t*tant-int1*tantdt" "` [integrating by parts] `=t*tant+log|cost|+C` `=(tan^(-1)x)*x+log|(1)/(sqrt(1+x^(2)))|+C` `[becausetant=xrArrcost=(1)/(sqrt(1+x^(2)))]` `=x(tan^(-1)x)-(1)/(2)log|1+x^(2)|+C`. (iii) Put `sec^(-1)x=t" so that"x=sectanddx=sec t tantdt`. `:.intsec^(-1)xdx=intt(sec t tan t)dt` `=t(sect)-int1*sect dt" "` [integrating by parts] `=t(sect)-log|sec t+tant|+C` `=t(sect)-log|sect+sqrt(sec^(2)t-1)|+C` `=x(sec^(-1)x)-log|x+sqrt(x^(2)-1)|+C`. |
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| 157. |
Evaluate the following integrals: `intsin^(3)xcosxdx` |
| Answer» Correct Answer - `(sin^(4)x)/(4)+C` | |
| 158. |
`inte^(x)((1+sinx)/(1+cosx))dx=?` |
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Answer» Correct Answer - `e^(x)"tan"(x)/(2)+C` `I=inte^(x){(1)/(2cos^(2)(x//2))+(2sin(x//2)cos(x//2))/(2cos^(2)(x//2))}dx` `=inte^(x)*{(1)/(2)"sec"^(2)(x)/(2)+"tan"(x)/(2)}dx=e^(x){"tan"(x)/(2)+(1)/(2)sec^(2)(x)/(2)}dx=e^(x)"tan"(x)/(2)+C`. |
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| 159. |
`int ((1+cotx))/((x+logsinx))dx.` |
| Answer» Correct Answer - `log|x+logsinx|+C` | |
| 160. |
Evaluate the following integrals: `intcos^(4)xsin^(3)xdx` |
| Answer» Correct Answer - `(1)/(7)cos^(7)x-(1)/(5)cos^(5)x+C` | |
| 161. |
Evaluate the following integrals: `intcos^(3)(3x+5)dx` |
| Answer» Correct Answer - `(1)/(3)sin(3x+5)-(1)/(9)sin^(3)(3x+5)+C` | |
| 162. |
Evaluate the following integrals: (i) `intcos^(2)(x//2)dx` (ii) `intcot^(2)(x//2)dx` |
| Answer» Correct Answer - (i) `(1)/(2)x+(1)/(2)sinx+C` (ii) `-2cot(x//2)-x+C` | |
| 163. |
Evaluate the following integrals: `inte^(2x)sinxdx` |
| Answer» Correct Answer - `(1)/(5)*e^(2x)(2sinx-cosx)+C` | |
| 164. |
Evaluate the following integrals: `inte^(2x)cos(3x+4)dx` |
| Answer» Correct Answer - `(e^(2x))/(13)*{2cos(3x+4)+3sin(3x+4)}+C` | |
| 165. |
`inte^(x)(cotx+logsinx)dx=?` |
| Answer» Correct Answer - `e^(x)log(sinx)+C` | |
| 166. |
Evaluate the following integrals: `inte^(-x)cosxdx` |
| Answer» Correct Answer - `(1)/(2)e^(-x)(sinx-cosx)+C` | |
| 167. |
Evaluate the following integrals: `inte^(x)(sinx+cosx)dx` |
| Answer» Correct Answer - `e^(x)sinx+C` | |
| 168. |
Evaluate the following integrals: `inte^(x)secx(1+tanx)dx` |
| Answer» Correct Answer - `e^(x)secx+C` | |
| 169. |
Evaluate the following integrals: `int(log(logx))/(x)dx` |
| Answer» Correct Answer - `(logx)[log(logx)-1]+C` | |
| 170. |
Evaluate the following integrals: `intcot^(-1)xdx` |
| Answer» Correct Answer - `xcot^(-1)x+(1)/(2)log(1+x^(2))+C` | |
| 171. |
Evaluate the following integrals: `inte^(x)(cotx-"cosec"^(2)x)dx` |
| Answer» Correct Answer - `e^(x)cotx+C` | |
| 172. |
Evaluate the following integrals: `intx^(2)cot^(-1)xdx` |
| Answer» Correct Answer - `(1)/(3)x^(3)cot^(-1)x+(1)/(6)x^(2)-(1)/(6)log|x^(2)+1|+C` | |
| 173. |
Evaluate the following integrals: `intcos^(-1)sqrt(x)dx` |
| Answer» Correct Answer - `(1)/(2)(2x-1)cos^(-1)sqrt(x)-(1)/(2)sqrt(x(1-x))+C` | |
| 174. |
Evaluate the following integrals: `intxcot^(-1)xdx` |
| Answer» Correct Answer - `(1)/(2)x^(2)(cot^(-1)x)+(1)/(2)(x-tan^(-1)x)+C` | |
| 175. |
Evaluate the following integrals: `intxcos^(-1)xdx` |
| Answer» Correct Answer - `(1)/(2)x^(2)cos^(-1)-(1)/(4)xsqrt(1-x^(2))+(1)/(4)sin^(-1)x+C` | |
| 176. |
Evaluate the following integrals: `intx^(2)e^(3x)dx` |
| Answer» Correct Answer - `e^(3x)((x^(2))/(3)-(2x)/(9)+(2)/(27))+C` | |
| 177. |
Evaluate the following integrals: `intx^(2)sin^(2)xdx` |
| Answer» Correct Answer - `(1)/(6)x^(3)-(1)/(4)sin2x-(1)/(4)x^(2)sin2x-(1)/(4)xcos2x+(1)/(8)sin2x+C` | |
| 178. |
Evaluate the following integrals: `int(logx)/(x^(n))dx` |
| Answer» Correct Answer - `(x^(1-n))/((1-n))*log|x|-(x^(1-n))/((1-n)^(2))+C` | |
| 179. |
Evaluate the following integrals: `intx^(2)e^(x)dx` |
| Answer» Correct Answer - `e^(x)(x^(2)-2x+2)+C` | |