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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. |
`(5x)/((x+1)(x^(2)+9))` |
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Answer» `int(5x)/((x+1)(x^(2)+9))dx` माना `(5x)/((x+1)(x^(2)+9))=(A)/(x+1)+(Bx+C)/(x^(2)+9)` `rArr" "5x=A(x^(2)+9)+(Bx+C)(x+1)` `x=-1` तो `-5=A(1+9)+0rArr A=-(1)/(2)` `x=0` तो `0=9A+C" "rArr" "C=(9)/(2)` `x^(2)` के गुणांकों को बराबर, `A+B=0` `rArr" "B=-A=(1)/(2)` `thereforeint(5x)/((x+1)(x^(2)+9))dx=int((-(1)/(2)))/((x+1))dx+int((1)/(2)x+(9)/(2))/(x^(2)+9)dx` `" "=-(1)/(2)int(1)/(x+1)dx+(1)/(2)int((x+9)/(x^(2)+9))dx` `" "=-(1)/(2)log|x+1|+(1)/(2).(1)/(2)int(2x)/(x^(2)+9)dx+(9)/(2)int(1)/(x^(2)+9)dx` `" "=-(1)/(2)log|x+1|+(1)/(4)log|x^(2)+9|+(9)/(2).(1)/(3)tan^(-1)((x)/(3))+C` `" "=-(1)/(2)log|x+1|+(1)/(4)log|x^(2)+9|+(3)/(2)tan^(-1)((x)/(3))+C` |
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| 52. |
`int_(0)^(pi//2) sin^(3)xdx` का मान ज्ञात कीजिए। |
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Answer» `int_(0)^(pi//2)sin^(3)xdx` `=int_(0)^(pi//2)(3sinx-sin3x)/(4)dx` `=(1)/(4)[-3cosx+(1)/(3)cos3x]_(0)^(pi//2)` `=(1)/(4)[-3cos.(pi)/(2)+(1)/(3)cos.(3pi)/(2)]-(1)/(4)[-3 cos 0+(1)/(3)cos0]` `=0-(1)/(4)(-3+(1)/(3)=(2)/(3))` |
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| 53. |
`int_(1)^(2)(cos(logx))/(x)dx` का मान ज्ञात कीजिए । |
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Answer» माना `logx=t" "rArr" "(1)/(x)dx=dt` `x = a" पर "t=log 1=0` `x = 2 " पर "t=log2` `therefore int_(1)^(2)(cos(logx))/(x)dx=int_(0)^(log2)cos tdt` `=[sint]_(0)^(log2)=sin(log2)-sin 0` `=sin(log2)` |
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| 54. |
`int_(0)^((pi)/(2))(sinx)/(1+cos^(2)x)dx` |
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Answer» माना `" "I=int_(0)^(pi//2)(sinx)/(1+cos^(2)x)dx` माना `" "cos x=t` `rArr -sin x dx=dt rArr sin x dx=- dt` `x =0" "rArr" "t=cos 0=1` और `" "x=(pi)/(2)" "rArr" "t=cos.(pi)/(2)=0` `therefore" "I=int_(1)^(0)(1)/(1+t^(2))(-dt)=-int_(1)^(0)(1)/(1+t^(2))dt` `=-[tan^(-1)t]_(1)^(0)` `=-[tan^(-1)(0)-tan^(-1)(1)]` `=-(0-(pi)/(4))=(pi)/(4)` |
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| 55. |
`int_(1)^(2)(dx)/(xsqrt(x^(2)-1))` का मान ज्ञात कीजिए। |
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Answer» `int_(1)^(2)(dx)/(xsqrt(x^(2)-1))` `=[sec^(-1)x]_(1)^(2)=sec^(-1)2-sec^(-1)1` `=(pi)/(3)-0=(pi)/(3)` |
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| 56. |
`(1)/(x^((1)/(2))+x^((1)/(3)))` संकेत `(1)/(x^((1)/(2))+x^((1)/(3)))=(1)/(x^((1)/(3))(1+x^((1)/(6)))), x=t^(6)` रखिए । |
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Answer» माना `I=int(1)/(x^(1//2)+x^(1//3))dx` माना `x=t^(6)" "rArr" "dx=6t^(5)dt` `therefore" "I=int(6t^(5))/((t^(6))^(1//2)+(t^(6))^(1//3))dt=6int(t^(5))/(t^(3)+t^(2))dt` `" "=6int(t^(5))/(t^(2)(t+1))dt=6int(t^(3))/((t+1))dt` `" "=6int((t^(3)+1-1)/(t+1))dt` `" "=6int[((t+1)(t^(2)-t+1))/(t+1)-(1)/(t+1)]dt` `" "=6int(t^(2)-t+1-(1)/(t+1))dt` `" "=6((t^(3))/(3)-(t^(2))/(2)+t-log|t+1|)+C` `" "=6{((x^(1//6))^(3))/(3)-((x^(1//6))^(2))/(2)+(x^(1//6))-log|x^(1//6)+1|+C}` `" "(because t=x^(1//6))` `" "=2x^(1//2)-3x^(1//3)+6x^(1//6)-6log|x^(1//6)+1|+c` |
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| 57. |
`(1)/(x(x^(n)+1))" "` [ संकेत : अंश और हर को `x^(n-1)` से गुणा कीजिए और `x^(n)=t` रखिए ] |
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Answer» माना `I=int(1)/(x(x^(n)+1))dx=int(x^(n-1))/(x^(n)(x^(n)+1)dx` `therefore" "I=int(dt)/(n.t(t+1))" माना "x^(n)=t` `" "=(1)/(n) int ((t+1)-t)/(t(t+1))dt" "rArr n.x^(n-1)dx=dt` `" "=(1)/(n)int((1)/(t)-(1)/(t+1))dt" "rArr x^(n-1)dx=(dt)/(n)` `" "=(1)/(n)[log|t|-log(t+1)]+C` `=(1)/(n)log|(t)/(t+1)|+C` `=(1)/(n)log|(x^(n))/(x^(n)+1)|+C` |
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| 58. |
`int_(0)^(2)xsqrt(x+2)dx (x+2=t^(2)" रखिए ")` |
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Answer» माना `x+2=t^(2)" "rArr" "dx=2t dt` `x=0` पर `t=sqrt(0+2)=sqrt2` `x=2" पर "t=sqrt(2+2)=2` `therefore " "int_(0)^(2)x sqrt(x+2)dx` `=int_(sqrt2)^(2)(t^(2)-2)sqrt(t^(2)).2t dt` `=2int_(sqrt2)^(2)(t^(2)-2t^(2))dt` `=2[(t^(5))/(5)-(2t^(3))/(3)]_(sqrt2)^(2)` `=(2)/(15)[3t^(5)-10t^(3)]_(sqrt2)^(2)` `=(2)/(15)[(96-80)-(12sqrt2-20sqrt2)]` `=(2)/(15)(16+8sqrt3)=(16)/(15)(2+sqrt2)` |
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| 59. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(0)^(1)xsqrt((1-x^(2))/(1+x^(2)))dx` |
| Answer» `((pi)/(4)-(1)/(2))` | |
| 60. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(1)^(2)(1)/(xsqrt(x^(2)-1))dx` |
| Answer» Correct Answer - `(pi)/(3)` | |
| 61. |
`(1)/(xsqrt(ax-x^(2)))` [संकेत : `x=(a)/(t)` रखिए] |
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Answer» माना`" "I=int(1)/(xsqrt(ax-x^(2)))dx` माना `x=(a)/(t)" "rArr" "dx=-(1)/(t^(2))dt` `therefore" "I=int(1)/((a)/(t)sqrt(a((a)/(t))-(a^(2))/(t^(2))))((-a)/(t^(2)))dt` `" "=int(-a)/(a.asqrt(t-1))dt=-(1)/(a)int(t-1)^(-1//2)dt` `" "=-(1)/(a).((t-1)^((-1//2)+1))/(-(1//2)+1)+C=-(2)/(a)sqrt(t-1)+C` `" "=-(2)/(a)sqrt((a)/(x)-1)+C=-(2)/(a)sqrt((a-x)/(x))+C` |
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| 62. |
`xsqrt(1+2x^(2))` |
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Answer» `intxsqrt(1+2x^(2))dx" माना "1+2x^(2)=t` `=intsqrtt (dt)/(4)=(1)/(4).(t^(3//2))/(3//2)+c" "rArr 4xdx=dt` `=(1)/(6)(1+2x^(2))^(3//2)+c" "rArr" "xdx =(dt)/(4)` |
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| 63. |
`(4x+2)sqrt(x^(2)+x+1)` |
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Answer» `int(4x+2)sqrt(x^(2)+x+1)dx" माना "x^(2)+x+1=t` `=2int(2x+1)sqrt(x^(2)+x+1)dx" "rArr (2x+1)dx=dt` `=2int(2x+1)sqrt(x^(2)+x+1)dx` `=2intsqrttdt=2.(t^(3//2))/(3//2)+c` `=(4)/(3)(x^(2)+x+1)^(3//2)+c` |
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| 64. |
`int(x^(3))/(x^(4)+2)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(x^(3))/(x^(4)+2)dx` माना `" "x^(4)+2=t` `rArr" "4x^(3)=(dt)/(dx)` `rArr" "x^(3)dx=(dt)/(4)` `therefore" "I=int(dt)/(4.t)=(1)/(4)log|t|+c` `" "=(1)/(4)log|x^(4)+2|+c` |
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| 65. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(sin^(-1)x)/((1-x^(2))^(3//2))dx` |
| Answer» `(xsin^(-1))/(sqrt(1-x^(2)))+(1)/(2)log(1-x^(2))+c` | |
| 66. |
`int(x^(x)-e^(-x))/(e^(x)+e^(-x))dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(e^(x)-e^(-x))/(e^(x)+e^(-x))dx` और `" "e^(x)+e^(-x)=t` `rArr" "(e^(x)-e^(-x))dx=dt` `therefore" "I=int(1)/(t).dt=log|t|+c` `" "=log|e^(x)+e^(-x)|+c` |
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| 67. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(0)^(pi//2)(dx)/(4sin^(2)x+5cos^(2)x)` |
| Answer» Correct Answer - `(pi)/(4sqrt5)` | |
| 68. |
`int sqrt((a+x)/(a-x))` का मान ज्ञात कीजिए । |
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Answer» `int sqrt((a+x)/(a-x))dx=intsqrt(((a+x)(a+x))/((a-x)(a+x)))dx` `" "=int(a+x)/(sqrt(a^(2)-x^(2)))` `=int(a)/(sqrt(a^(2)-x^(2)))dx+int(x)/(sqrt(a^(2)-x^(2)))dx` माना `a^(2)-x^(2)=t` `=a int(1)/(sqrt(a^(2)-x^(2)))dx+int(dt)/(-2sqrtt)" "{:(rArr" "-2x=(dt)/(dx)),(rArr" "xdx=(dt)/(dx)):}` `" "=a.sin^(-1).(x)/(a)-(1)/(2).2sqrtt+c` `" "=asin^(-1).(x)/(a)-sqrt(a^(2)-x^(2))+c` |
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| 69. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(x^(2)tan^(-1)x)/(1+x^(2))dx` |
| Answer» `x tan^(-1)x-(1)/(2)log(1+x^(2))-(1)/(2)(tan^(-1)x)^(2)+c` | |
| 70. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(0)^(1)(1)/(e^(x)+e^(-x))dx` |
| Answer» `tan^(-1)e-(pi)/(4)` | |
| 71. |
`int(sin(x-a))/(sin(x+a))dx` का मान ज्ञात कीजिए । |
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Answer» माना `" "I=int(sin(x-a))/(sin(x+a))dx` और `" "x+a=t` `rArr" "dx=dt` `therefore" "=int(sin(t-2a))/(sint)dt` `" "=int(sint.cos2a-costsin2a)/(sint)dt` `" "=intcos2adt-intsin2a.cott dt` `" "=tcos2a-sin2a.log|sint|+c` `" "=(x+a)cos2a-sin2a log|sin(x+a)|+c` `" "=xcos2a-sin2alog|sin(x+a)|c_(1)` `" "`(जहाँ `c_(1)=c+acos2a` है ) |
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| 72. |
`int(dx)/(2+cosx)` का मान ज्ञात कीजिए । |
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Answer» `int(1)/(2+cosx)dx` `" "int(1)/(2+2cos^(2).(x)/(2)-1)dx=int(1)/(1+2cos^(2).(x)/(2))dx` `" "=int(1)/(sin^(2).(x)/(2)+cos^(2).(x)/(2)+2cos^(2).(x)/(2))dx` `" "=int(1)/(sin^(2).(x)/(2)+3cos^(2).(x)/(2))dx` `" "=int(sec^(2).(x)/(2))/(tan^(2).(x)/(2)+3)dx" माना "tan.(x)/(2)=t` `=int(2dt)/(t^(2)+3)" "{:(rArr(1)/(2)sec^(2).(x)/(2)=(dt)/(dx)),(rArrsec^(2).(x)/(2)dx=2dt):}` `" "=2int(1)/(t^(2)+(sqrt3)^(2))dt` `" "=(2)/(sqrt3)tan^(-1)((t)/(sqrt3))+c` `" "=(2)/(sqrt3)tan^(-1)((tan.(x)/(2))/(sqrt3))+c` |
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| 73. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int tan^(-1)((3x-x^(3))/(1-3x^(2)))dx` |
| Answer» `3xtan^(-1)x-(3)/(2)log(1+x^(2))+c` | |
| 74. |
`int(sin2x)/(acos^(2)x+bsin^(2)x)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int((sin2x)/(acos^(2)x+bsin^(2)x))dx` और `" "acos^(2)x+bsin^(2)x=t` `rArr" "a.2cosx(-sinx)+b.2sinx.cosx=(dt)/(dx)` `rArr" "sin2x(b-1)dx=dt` `rArr" "sin2x.dx=(dt)/(b-a)` `therefore" "I=int(dt)/((b-a).t)=(1)/(b-a).log|t|c` `" "=(1)/(b-a)log|acos^(2)x+bsin^(2)x|+c` |
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| 75. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(1)^(2)(dx)/(x(1+logx)^(2))` |
| Answer» Correct Answer - `(log2)/(1+log2)` | |
| 76. |
निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(1)^(3)(cos(logx))/(x)dx` |
| Answer» Correct Answer - `sin(log3)` | |
| 77. |
`int(1)/(sqrt(1-e^(2x)))` का मान ज्ञात कीजिए । |
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Answer» `int(1)/(sqrt(1-e^(2x)))dx` `" "=int(1)/(sqrt(e^(2x)(e^(-2x)-1)))dx=int(1)/(e^(x)sqrt(e^(-2x)-1))dx` `" "=int(e^(-x))/(sqrt((e^(-x))^(2)-1))dx" माना "e^(-x)=t` `" "=int(-dt)/(sqrt(t^(2)-1))" "{:(rArr-e^(-x)dx=dt),(rArre^(-x)dx=-dt):}` `" "=log|t+sqrt(t^(2)-1)|+c` `" "=-log|e^(-x)+sqrt(e^(-2x)-1)|+c` |
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| 78. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(x^(2))/((x^(2)+1)(x^(2)+4))dx` |
| Answer» `-(1)/(3)tan^(-1)x+(2)/(3)tan^(-1).(x)/(2)+c` | |
| 79. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int cos^(-1)((1-x^(2))/(1+x^(2)))dx` |
| Answer» `2x tan^(-1)x-log(1+x^(2))+c` | |
| 80. |
`int(1)/(e^(x)-1)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(1)/(e^(x)-1)dx=int(e^(-x))/(e^(-x)(e^(x)-1))dx` `" "=int(e^(-x))/(1-e^(-x))dx` माना`1-e^(-x)=t` `rArr" "e^(-x)=(dt)/(dx)` `rArr" "e^(-x)dx=dt` `therefore" "I=int(dt)/(t)=log|t|+c` `" "=log|1-e^(-x)|+c` |
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| 81. |
`int(1)/(4_x^(2))dx` का मान ज्ञात कीजिए । |
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Answer» `int(1)/(4+x^(2))dx` `" "=int(1)/(x^(2)+2^(2))dx=(1)/(2)tan^(-1).(x)/(2)+c` |
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| 82. |
निम्नलिखित समाकलों को हल कीजिए । `int(x^(3))/((4-x^(4))^(2))dx` |
| Answer» `(1)/(4(4-x^(4)))+c` | |
| 83. |
निम्नलिखित समाकलों को हल कीजिए । `int(cos^(2)(logx))/(x)dx` |
| Answer» `(1)/(2)logx+(1)/(4)sin(2logx)+c` | |
| 84. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(x+3)/(x^(2)-9)dx` |
| Answer» `x+(sqrt3+1)/(2)log|x-sqrt3|-(sqrt3-1)/(2)log|x+sqrt3|log|x|+c` | |
| 85. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int sin^(-1)((2x)/(1+x^(2)))dx` |
| Answer» `2[x tan^(-1)x-(1)/(2)log(1+x^(2))]+c` | |
| 86. |
निम्नलिखित समाकलों को हल कीजिए । `int(x^(2))/(sqrt(1+x^(3)))dx` |
| Answer» `(2)/(3)sqrt(1+x^(3))+c` | |
| 87. |
`intcos 2x cos 4x cos 6xdx` |
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Answer» `int cos 2x cos 4x cos 6x dx` `=(1)/(2) int cos 4x(2 cos 6x cos 2x)dx` `=(1)/(2) int cos 4x(cos 8x + cos 4x)dx` `=(1)/(4) int(2 cos 8x cos 4x+2 cos^(2)4x)dx` `=(1)/(4) int (cos 12x+cos 4x+cos 8x+1)dx` `=(1)/(4)((sin12x)/(12)+(sin 4x)/(4)+(sin8x)/(8)+c)+c` |
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| 88. |
`int(1-tanx)/(1+tanx)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(1-tanx)/(1+tanx)dx=int(cosx-sinx)/(cosx+sinx)dx` माना `cosx+sinx+t` `rArr" "-sinx+cosx=(dt)/(dx)` `rArr" "(cosx-sinx)dx=dt` `therefore" "I=int(1)/(t)dt=log|t|+c` `" "=log|cosx+sinx|+c` |
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| 89. |
निम्नलिखित समाकलों को हल कीजिए । `int(1)/(x.cos^(2)(log_(e)x))dx` |
| Answer» Correct Answer - `tan(log_(e)x)+c` | |
| 90. |
`int(1)/(cos(x+a).sin(x+b))dx` का मान ज्ञात कीजिए । |
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Answer» `int(1)/(cos(x+a).sin(x+b))dx` `" "=(1)/(cos(a-b))int(cos(a-b))/(cos(x+a).sin(x+b))dx` `" "=(1)/(cos(a-b))int(cos{(x+a)-(x+b)})/(cos(x+a).sin(x+b))dx` `" "=(1)/(cos(a-b))int(+sin(x+a)sin(x+b))/(cos(x+a)sin(x+b))dx` `" "=(1)/(cos(a-b))int{(cos(x+a)cos(x+b))/(cos(x+a)sin(x+b))+(sin(x+a)sin(x+b))/(cos(x+a)sin(x+b))}dx` `" "=(1)/(cos(a-b))int{cot(x+b)+tan(x+a)}dx` `" "=(1)/(cos(a-b))[log|sin(x+b)|+log|sec(x+a)|]+c` |
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| 91. |
मान ज्ञात कीजिए - `int cos3x cos 4x dx` |
| Answer» `(sin7x)/(14)+(sinx)/(2)` | |
| 92. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `int(dx)/(x(1+log_(e)x)(3+log_(e)x))` |
| Answer» `(1)/(2)log|(1+logx)/(3+logx)|+c` | |
| 93. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `intx^(3).e^(x^(2))dx` |
| Answer» `(1)/(2)e^(x^(2))(x^(2)-1)+c` | |
| 94. |
`sin^(3)(2x+1)` |
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Answer» `intsin^(3)(2x+1)dx` `=int sin^(2)(2x+1)sin(2x+1)dx` `=int(1-cos^(2)(2x+1)}sin(2x+1)dx` `=int(1-t^(2))(dt)/(-2)=-(1)/(2)[t-(t^(3))/(3)]+c` माना `cos(2x+1)=t` `rArr -2 sin (2x+1)dx=dt` `rArr sin(2x+1)dx=(dt)/(2)` `=-(1)/(2)cos(2x+1)+(1)/(6)cos^(3)(2x+1)+c` |
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| 95. |
निम्नलिखित समाकलों को हल कीजिए । `int((log_(e)x)^(3))/(x)dx` |
| Answer» `(1)/(4)(log_(e)x)^(4)+c` | |
| 96. |
`int(tan(sin^(-1)x)/(sqrt(1-x^(2))))dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(tan(sin^(-1)x))/(sqrt(1-x^(2)))dx` माना `sin^(-1)x=t` `rArr" "(1)/(sqrt(1-x^(2)))=(dt)/(dx)` `rArr" "(1)/(sqrt(1-x^(2)))dx=dt` `therefore" "I=inttant.dt=log|sect|+c` `" "=log|sec(sin^(-1)x)|+c` |
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| 97. |
`int(cot(logx))/(x)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(cot(logx))/(x)dx` माना `" "logx=t` `rArr" "(1)/(x)=(dt)/(dx)` `rArr" "(1)/(x)dx=dt` `therefore" "I=intcot tdt` `" "=log|sint|+c` `" "=log|sin(logx)|+c` |
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| 98. |
मान ज्ञात कीजिए - `int cosx cos3x dx ` |
| Answer» `(1)/(8)sin4x+(1)/(4)sin2x` | |
| 99. |
निम्नलिखित समाकलनों के मान ज्ञात कीजिए- `intcot^(-1)xdx` |
| Answer» `x cot^(-1)x+(1)/(2)log(1+x^(2))+c` | |
| 100. |
`int(10x^(9)+10^(x)log_(e)10dx)/(x^(10)+10^(x))` बराबर है :A. `10^(x)-x^(10)+C`B. `10^(x)+x^(10)+C`C. `(10^(x)-x^(10))^(-1)+C`D. `log(10^(x)+x^(10))+C` |
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Answer» Correct Answer - D `int[(10x^(9)+10^(x)loge^(10))/(x^(10)+10^(x))]dx" "{:(" माना "x^(10)+10^(x)=t),(rArr(10x^(9)+10^(x)log_(e)10)),(" "dx=dt):}` `=int(1)/(t)dt=log|t|+c` `=log|x^(10)+10^()|+c` |
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