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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. |
निम्नलिखित समाकलों को हल कीजिए । `int(sinx.cosx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)` |
| Answer» `(1)/(2(b^(2)-a^(2)))log_(e)(a^(2)cos^(2)x+b^(2)sin^(2)x)+c` | |
| 152. |
`intsin^(4)xdx` |
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Answer» `intsin^(4)x dx =int(sin^(2)x)^(2)dx` `=int((1-cos2x)/(2))^(2)dx` `=(1)/(4)int(1-2cos2x+cos^(2)2x)dx` `=(1)/(4)int(1-2 cos 2x+(1+cos 4x)/(2))dx` `=(1)/(8)int(3-4cos 2x+cos 4x)/(2))dx` `=(1)/(8)[3x-(4sin2x)/(2)+(sin 4x)/(4)]+c` `=(3x)/(8)-(1)/(4)sin 2x+(1)/(32)sin 4x+c` |
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| 153. |
`int cosecx(cosecx+cotx)dx `का मान ज्ञात कीजिए | |
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Answer» `int cosecx (cosecx+cotx)dx` `=intcosec^(2)xdx+intcosecx.cotxdx` `=-cotx-cosecx+c` |
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| 154. |
`int (sin^(6)x+cos^(6)x)/(sin^(2)xcos^(2)x)dx` का मान ज्ञात कीजिए | |
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Answer» `int(sin^(6)x+cos^(6)x)/(sin^(2)xcos^(2)x)dx` `=int((sin^(2)x+cos^(2)x)^(3)-3sin^(2)xcos^(2)x(sin^(2)x+cos^(2)x))/(sin^(2)xcos^(2)x)dx` `=int(1-3sin^(2)xcos^(2)x)/(sin^(2)xcos^(2)x)dx` `=int((1)/(sin^(2)xcos^(2)x)-3)dx=int((sin^(2)x+cos^(2)x)/(sin^(2)xcos^(2)x-3))dx` `=int(sin^(2))/(sin^(2)xcos^(2)x)dx+int(cos^(2)x)/(sin^(2)xcos^(2)x)dx-3intdx` `=int(1)/(cos^(2)x)dx+int(1)/(sin^(2)x)dx-3intdx` `=intsec^(2)xdx+intcosec^(2)xdx-3intdx` `=tanx-cotx-3x+c` |
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| 155. |
`int(x)/(1+cosx)dx` का मान है -A. `x tan.(x)/(2)-2 log cos.(x)/(2)+c`B. `x tan.(x)/(2)+2 log cos.(x)/(2+c`C. `x tan.(x)/(2)+2log sin.(x)/(2)+c`D. इनमे से कोई नहीं |
| Answer» Correct Answer - B | |
| 156. |
`int(x - sinx)/(1-cosx)dx` का मान ज्ञात कीजिए । |
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Answer» `int(x-sinx)/(1-cosx)dx=int(x-2sin.(x)/(2)cos.(x)/(2))/(2sin^(2).(x)/(2))dx` `=int(x)/(2sin^(2).(x)/(2))dx-int(2sin.(x)/(2)cos.(x)/(2))/(2sin^(2).(x)/(2))dx` `=(1)/(2)int x" cosec"^(2)(x)/(2)dx- intcot.(x)/(2)dx` `=(1)/(2)[x int" cosec"^(2)(x)/(2)dx-int{(d)/(dx)x int" cosec"^(2)(x)/(2)dx}dx]-intcot.(x)/(2)dx` `=(1)/(2)[x.((-cot.(x)/(2)))/((1)/(2))-int1.((-cot.(x)/(2)))/((1)/(2))dx]-intcot.(x)/(2)+c` `=-x cot.(x)/(2)+intcot.(x)/(2)dx- int cot.(x)/(2)dx+c` `=-x cot.(x)/(2)+c` |
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| 157. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1)/((x+b)(x^(2)+a^(2))dx` |
| Answer» `(1)/(a^(2)+b^(2)).[log.(x+b)/(sqrt(x^(2)+a^(2)))+(b)/(a)tan^(-1).(x)/(a)]+c` | |
| 158. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1)/(x^(2)+2x+5)dx` |
| Answer» `(1)/(2)tan^(-1)((x+1)/(2))+c` | |
| 159. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1)/(sqrt(9-4x^(2)))dx` |
| Answer» `(1)/(2)sin^(-1).(2x)/(3)+c` | |
| 160. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(cosx)/(cos3x)dx` |
| Answer» `(1)/(2sqrt3)log|(1+sqrt3tanx)/(1-sqrt3 tanx)|+c` | |
| 161. |
निम्नलिखित समाकलों को हल कीजिए । `int(a)/(b+c.e^(x))dx` |
| Answer» `-(a)/(b).log|b.e^(-x)+c|+c_(1)` | |
| 162. |
निम्नलिखित समाकलों को हल कीजिए । `int(sin2x)/(5-cos^(2)x)` |
| Answer» Correct Answer - `log(5-cos^(2)x)+c` | |
| 163. |
निम्नलिखित समाकलों को हल कीजिए । (i) `int(1)/(sqrt(1-x^(2)).cos^(-1)x)dx` (ii) `int(sin(tan^(-1)x))/(1+x^(2))dx` |
| Answer» (i) `-log|cos^(-1)x|+c` (ii) `-cos(tan^(-1)x)+c` | |
| 164. |
निम्नलिखित समाकलों को हल कीजिए । `int(sinx)/(sin(x-1))dx` |
| Answer» `x cos a+sina log|sin(x-a)|+c` | |
| 165. |
निम्नलिखित समाकलों को हल कीजिए । `int(1+cosx)/(1+x^(2))dx` |
| Answer» (i) `log|x cosx|+c` (ii) `(1)/(-2(x+sinx)^(2))+c` | |
| 166. |
`intsin^(3)x cos^(3)xdx` का मान है -A. `(1)/(4)sin^(4)x-(1)/(8)sin^(6)x+c`B. `(1)/(4)sin^(4)x-(1)/(6)sin^(6)x+c`C. `(1)/(4)sin^(4)x+(1)/(6)sin^(6)x+c`D. इनमे से कोई नहीं |
| Answer» Correct Answer - B | |
| 167. |
`int ((1-sinx))/(cos^(2)x)dx ` का मान ज्ञात कीजिए | |
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Answer» `int((1-sinx))/(cos^(2)x)dx=int(1)/(cos^(2)x)dx-int(sinx)/(cos^(2)x)dx` `=intsec^(2)xdx -inttanxsecxdx` `=tanx-secx+c` |
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| 168. |
`inte^(x)cos^(2)x(cosx-3 sinx)dx` का मान ज्ञात कीजिए । |
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Answer» `inte^(x)cos^(2)x(cosx-3sinx)dx` `=inte^(x).cos^(3)xdx - inte^(x).3 cos^(2)x sin x dx` `=cos^(3)x inte^(x)dx- int{((d)/(dx).cos^(3)x)inte^(x)dx}dx- int e^(x).3 cos^(2)x sin x dx` `=e^(x).cos^(3)x - int3 cos^(2)x(-sinx).e^(x)dx- inte^(x).3 cos^(2)x sin x dx +c` `=e^(x) cos^(3)x+int3 cos^(2)x sin x e^(x)dx- int 3 cos^(2)x sin x e^(x)dx+c` `=e^(x)cos^(3)x+c` |
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| 169. |
योगफल की सीमा के रूप में निम्नलिखित समाकलनों के मान ज्ञात कीजिए - `int_(1)^(b)sinxdx` |
| Answer» Correct Answer - `cosa- cos b` | |
| 170. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `intsec^(6//5)x."cosec"^(4//5)xdx` |
| Answer» Correct Answer - `5(tanx)^(1//5)+c` | |
| 171. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1)/(cos(x-a)cos(x-b))dx` |
| Answer» `(1)/(sin(a-b))log|(cos(x-a))/(cos(x-b))|+c` | |
| 172. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1-cosx)/(cosx(1+cosx))dx` |
| Answer» `log|secx+tanx|-2tan.(x)/(2)+c` | |
| 173. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `intxlog(1+x)dx` |
| Answer» `(x^(2)-1)/(2)log(1+x)-(x^(2))/(4)+(x)/(2)+c` | |
| 174. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(x-2)/(x^(3)).e^(x)dx` |
| Answer» Correct Answer - `(e^(x))/(x^(2))+c` | |
| 175. |
निम्नलिखित समाकलों को हल कीजिए । `int(x^(e-1)-e^(x-1))/(x^(e)-e^(x))dx` |
| Answer» `(1)/(e)log(x^(e)-e^(x))+c` | |
| 176. |
मान ज्ञात कीजिए - `int(sec^(2)x)/(cosec^(2)x)dx` |
| Answer» Correct Answer - `tanx-x+c` | |
| 177. |
`int(sin(x-alpha))/(sin(x+alpha))dx` का मान है -A. `x sin 2acosa+cos 2 alpha|sin(x+alpha)|+c`B. `x cos 2a+sin 2alphalog|sin(x+alpha)|+c`C. `xcos 2alpha-sin 2alpha log|sin(x+alpha)|+c`D. इनमे से कोई नहीं |
| Answer» Correct Answer - C | |
| 178. |
`int (2-3sinx)/(cos^(2)x)dx` का मान होगा -A. `2tanx+3secx+c`B. `2tanx-3secx+c`C. `secx-tanx+c`D. `sec^(2)x+c` |
| Answer» Correct Answer - B | |
| 179. |
`int(sec^(2)x)/("cosec"^(2)x)dx` |
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Answer» `int(sec^(2)x)/("cosec"^(2)x)dx=int(sin^(2)x)/(cos^(2)x)dx` `=int tan^(2) xdx= int (sec^(2)x-1)dx` `=intsec^(2)x dx - int 1.dx` `=tan x-x+c` |
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| 180. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int"cosec"^(3)xdx` |
| Answer» `-(1)/(2)"cosec" x cot x+(1)/(2)log|tan.(x)/(2)|+c` | |
| 181. |
`int (1-sinx)/(cos^(2)x)dx` का मान होगा -A. `tanx-secx+c`B. `tanx+secx+c`C. `cotx+cosecx+c`D. `cotx-cosecx+c` |
| Answer» Correct Answer - A | |
| 182. |
`inte^(x)(sinx+cosx)dx` का मान है -A. `e^(x)sinx+c`B. `e^(x)cosx+c`C. `-e^(x)sinx+c`D. `-e^(x)cosx+c` |
| Answer» Correct Answer - A | |
| 183. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- (i) `intsec^(7)x.sinxdx` (ii) `int(1)/(sinx.cos^(2)x)dx` |
| Answer» (i) `(1)/(6)sec^(6)x+c` (ii) `secx+log|"cosec x"-cot x|+c` | |
| 184. |
`(3x+5)/(x^(3)-x^(2)-x+1)` |
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Answer» `int(3x+5)/(x^(3)-x^(2)-x+1)dx` `" "=int(3x+5)/((x^(2)1)(x-1))dx` `" "=int(3x+5)/((x-1)(x+1)(x-1))dx` `" "=int(3x+5)/((x-1)^(2)(x+1))dx` माना `" "(3x+5)/((x-1)^(2)(x+1))=(A)/(x-1)+(B)/((x-1)^(2))+(C)/(x+1)` `rArr" "(3x+5)/((x-1)^(2)(x+1))` `" "=(A(x-1)(x+1)+B(x+1)+C(x-1)^(2))/((x-1)^(2)(x+1))` `rArr 3x+5=A(x-1)(x+1)+B(x+1)+C(x-1)^(2)` `{:(x=1,"तो ",3+5=0+B(2)+0,rArr.,B=4),(x=-1,"तो ",-3+5=0+0+C(-2)^(2),rArr.,C=(1)/(2)):}` `rArr" "x^(2)` के गुणकों को समान रखने पर, `" "0=A+C` `rArr A=-C=-(1)/(2)` `therefore int(3x+5)/((x-1)^(2)(x+1))dx` `" "=int(_(1)/(2))/(x-1)+(4)/((x-1)^(2))+((1)/(2))/(x+1)dx` `" "=-(1)/(2)int(1)/(x-1)dx+4int(1)/((x-1)^(2))dx+(1)/(2)int(1)/(x+1)dx` `=-(1)/(2)log|x-1|+4((x-1)^(-2+1))/((-2+1))+(1)/(2)log|x+1|+C` `=(1)/(2)log|(x+1)/(x-1)|-(4)/(x-1)+C` |
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| 185. |
`(2x-3)/((x^(2)-1)(2x+3))` |
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Answer» `int(2x-3)/((x^(2)-1)(2x+3))dx` `" "=int(2x-3)/((x-1)(x+1)(2x+3))dx` माना `" "(2x-3)/((x-1)(x+1)(2x+3))=(A)/((x-1))+(B)/((x+1))+(C)/((2x+3))` `rArr" "(2x-3)/((x-1)(x+1)(2x+3))` `" "(A(2x+3)(x+1)+B(x-1)(2x+3)+C(x-1)(x+1))/((x-1)(x+1)(2x+3))` `rArr 2x-3=A(2x+3)(x+1)+B(x-1)(2x+3)+C(x-1)(x+1)` `{:(x=1,"तो ",2-3=A(5)(2)+0+0,rArr.,A=-(1)/(10)),(x=-1,"तो ",-2-3=0+B(-2)(1)+0,rArr.,B=(5)/(2)):}` `rArr x=-(3)/(2)" तो "-3-3=0+0+C(-(5)/(2))(-(1)/(2))` `rArr C=-(24)/(5)` `therefore" "A=-(1)/(10), B=(5)/(2)" तथा "C=-(24)/(5)` `therefore" "int(2x-3)/((x^(2)-1)(2x+3))dx` `" "=int((-1))/(10(x-1))dx+(5)/(2)int(1)/(x+1)dx-(24)/(5)int(1)/(2x+3)dx` `=-(1)/(10)log|x-1|+(5)/(2)log|x+1|-(24)/(5)(log|2x+3|)/(2)+C_(1)` `=(5)/(2)log|x+1|-(1)/(10)log|x-1|-(12)/(5)log|2x+3|+C_(1)` |
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| 186. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(logx)^(2)dx` |
| Answer» `x(logx)^(2)-2xlogx+2x+c` | |
| 187. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(tan(1+logx))/(x)dx` |
| Answer» `log_(e)|sec(1+logx)|+c` | |
| 188. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(logx)/((1+logx)^(2))dx` |
| Answer» Correct Answer - `(x)/(1+logx)+c` | |
| 189. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(sinx cosx-1)/(sin^(2)x).e^(x)dx` |
| Answer» Correct Answer - `e^(x).cotx+c` | |
| 190. |
`int (sec^(2)x)/(cosec^(2)x)dx` का मान होगा -A. `tanx-x+c`B. `tanx+x+c`C. `tanx+2x+c`D. `secx+2x+c` |
| Answer» Correct Answer - A | |
| 191. |
`int(e^(m tan^(-1)x))/(1+x^(2))dx` का मान है -A. `(1)/(m)tan^(-1)x+c`B. `(1)/(m)e^(m tan^(-1)x)+c`C. `e^(m tan^(-1)x)+c`D. इनमे से कोई नहीं |
| Answer» Correct Answer - B | |
| 192. |
`inttan^(3)2x.sec 2xdx` का मान है -A. `(1)/(2)sec^(3)2x-(1)/(2)sec2x+c`B. `(1)/(2)sec^(3)2x+(1)/(2)sec2x+c`C. `(1)/(6)sec^(3)2x-(1)/(2)sec2x+c`D. `(1)/(6)sec^(3)2x+(1)/(2)sec2x+c` |
| Answer» Correct Answer - C | |
| 193. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(1)/(sqrt(1+cos2x))dx` |
| Answer» `(1)/(sqrt2)log|secx+tanx|+c` | |
| 194. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `inte^(3x)cos2x dx` |
| Answer» `(e^(3x))/(13)(2sinx-secx)+log(1+sinx)+c` | |
| 195. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int[sin(logx)+cos(logx)]dx` |
| Answer» Correct Answer - `x sin (logx)+c` | |
| 196. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(e^(x)(x^(2)+1))/((x+1)^(2))dx` |
| Answer» `(e^(x)(x-1))/(x+1)+c` | |
| 197. |
`(x^(2)+1)logx` |
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Answer» `int(x^(2)+1)log x dx` `=log x. int (x^(2)+1)dx` `" "-int{(d)/(dx)logx int(x^(2)+1)dx}dx` `=logx.((x^(3))/(3)+x)-int(1)/(x)((x^(3))/(3)+x)dx+C` `=(x+(x^(3))/(3))logx-int((x^(2))/(3)+1)dx+C` `=(x+(x^(3))/(3))logx-(x+(x^(3))/(9))+C` |
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| 198. |
`x sec^(2)x` |
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Answer» माना `int x sec^(2)x dx` `=x int sec^(2)xdx- int {(d)/(dx)x int sec^(2)xdx}dx` `=x tan x - int 1. tan x dx` `=x tan x - log |sec x|+c` |
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| 199. |
`tan^(-1)x` |
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Answer» `int tan^(-1)xdx` `=int tan^(-1)x.1dx` `=tan^(-1)x. int 1dx - int{(d)/(dx)tan%(-1)x. int 1dx}dx` ( `tan^(-1) x` को प्रथम लेने पर ) `=x tan^(-1)x-int(x)/(1+x^(2))dx" माना "1+x^(2)=t` `=x tan^(-1)x-int(dt)/(2t)" "therefore " "2x=(dt)/(dx)` `=x tan^(-1)x-(1)/(2)logt+C" "rArr" "xdx=(dt)/(2)` `=x tan^(-1)x-(1)/(2)log(1+x^(2))+C` |
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| 200. |
`int_(0)^((pi)/(2))cos 2x dx` |
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Answer» `int_(0)^(pi//2)cos 2x dx=[(sin2x)/(2)]_(0)^(pi//2)` `" "=(1)/(2)[sin2x]_(0)^(pi//2)` `" "=(1)/(2)[(sin2xx(pi)/(2))-sin(0)]` `" "=(1)/(2)(0-0)=0` |
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