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401.

`int (1)/(x(logx)^(m) )dx,x gt 0, m ne 1`

Answer» `int(1)/(x(logx)^(m))dx" माना "logx=t`
`=int(1)/(t^(m))dt=intt^(-m)dt" "rArr (1)/(x)dx=dt`
`=(t^(-m+1))/(-m+1)+c=((logx)^(1-m))/(1-m)+c`
Discussion
402.

`int(x-1)/((x+1)(x^(2)+1))dx` का मान ज्ञात कीजिए ।

Answer» माना `(x-1)/((x+1)(x^(2)+1))`
`" "=(A)/(x+1)+(Bx+C)/(x^(2)+1)`
`" "=(A(x^(2)+1)+(Bx+C)(x+1))/((x+1)(x^(2)+1))`
`rArr" "A(x^(2)+1)+(Bx+C)(x+1)=x-1`
समान घातों के गुणांकों की तुलना करने पर
`A+B=0`
`B+C=1`
`A+C=-1`
हल करने पर `A=-1,B=1, C=0`
`therefore int(x-1)/((x+1)(x^(2)+1))dx=int(-1)/(x+1)dx+int(x)/(x^(2)+1)dx`
`=int(-1)/(x+1)dx+(1)/(2) int (2x)/(x^(2)+1)dx`
`=-log|x+1|+(1)/(2)log|x^(2)+1|+c`
Discussion
403.

`int_(0)^(1)|5x-3|dx` का मान है -A. `(13)/(10)`B. `-(13)/(10)`C. `(3)/(10)`D. `-(3)/(10)`

Answer» Correct Answer - A
Discussion
404.

`int(dx)/(2x^(2)+x-1)` का मान ज्ञात कीजिए ।

Answer» `int(1)/(2x^(2)+x-1)dx=(1)/(2)(1)/(x^(2)+(1)/(2)x-(1)/(2))dx`
`=(1)/(2)int(1)/((x^(2)+(1)/(2)x+(1)/(16))-((1)/(2)+(1)/(16)))dx`
`=(1)/(2)int(1)/((x+(1)/(4))^(2)-((3)/(4))^(2))dx`
`=(1)/(2).(1)/(2.(3)/(4))log|(x+(1)/(4)-(3)/(4))/(x+(1)/(4)+(3)/(4))|+c`
`=(1)/(3)log|(2x-1)/(2x+2)|+c`
Discussion
405.

सिद्ध कीजिए कि - `int(1)/(x[6(logx)^(2)+7 logx+2])dx=log|(1+logx^(2))/(2+logx^(3))|+c`

Answer» माना `logx=t`
`rArr" "(1)/(x)dx=dt`
माना `" "I=int(1)/(x[6(logx)^(2)+7logx+2])dx`
`" "=int(1)/(6t^(2)+7t+2)dx`
`" "=int(1)/(6t^(2)+7t+2)dx`
`" "=int(1)/((3t+2)(2t+1))dt`
पुनः माना `(1)/((3t+2)(2t+1))=(A)/(3t+2)+(B)/(2t+1)=(A(2t+1)+B(3t+2))/((3t+2)(2t+1))`
`rArr" "A(2t+1)+B(3t+2)=1`
समान घातों के गुणांकों को बराबर रखने पर
`2A+3B=0`
`A+2B=1`
हल करने पर `B=2, A=-3`
`therefore(1)/((3t+2)(2t+1))=(-3)/(3t+2)+(2)/(2t+1)`
और `I=int(-3)/(3t+2)dt+int(2)/(2t+1)dt`
`=-log|3t+2|+log|2t+1|+c`
`=log|(2t+1)/(3t+2)|+c=log|(1+2logx)/(2+3logx)|+c`
`=log|(1+logx^(2))/(2+logx^(3))|+c`
Discussion
406.

`int_(0)^(pi)|cosx|dx` का मान ज्ञात कीजिए।

Answer» `|cosx|={{:(cosx",",0lexlepi//2),(-cosx",",(pi)/(2)lexlepi):}`
`therefore" "int_(0)^(pi)|cosx|dx=int_(0)^(pi//2)|cosx|dx+int_(pi//2)^(pi)|cosx|dx`
`=int_(0)^(pi//2)cosx dx-int_(pi//2)^(pi)cosx dx`
`=[sinx]_(0)^(pi//2)-[sinx]_(pi//2)^(pi)`
`=(sin.(pi)/(2)-sin0)-(sinpi-sin.(pi)/(2))`
`=1-0-(0-1)=2`
Discussion
407.

`int_(0)^(1)|5x-3|dx` का मान ज्ञात कीजिए।

Answer» `|5x-3|={{:(5x-3",",xle3//5),(-(5x-3)",",xlt3//5):}`
`therefore" "int_(0)^(1)|5x-3|dx=int_(0)^(3//5)|5x-3|dx+int_(3//5)^(1)|5x-3|dx`
`=int_(0)^(3//5)-(5x-3)dx+int_(3//5)^(1)(5x-3)dx`
`=-[(5x^(2))/(2)-3x]_(0)^(3//5)+[(5x^(2))/(2)-3x]_(3//5)^(1)`
`=-[(5)/(2)xx(9)/(25)-3xx(3)/(5)-0]+[((5)/(2)-3)-((5)/(2)xx(9)/(25)-3xx(3)/(5))]`
`=-(9)/(10)+(9)/(5)-(1)/(2)-(9)/(10)+(9)/(5)`
`=-(18)/(10)+(18)/(5)-(1)/(2)=(-18+36-5)/(10)`
`=(13)/(10)`
Discussion
408.

`int(1)/(1+x^(3))` का मान ज्ञात कीजिए।

Answer» माना `(1)/(1+x^(3))=(1)/((x+1)(x^(2)-x+1))`
`" "=(A)/(x+1)+(Bx+C)/(x^(2)-x+1)`
`" "=(A(x^(2)-x+1)+(Bx+C)(x+1))/((x+1)(x^(2)-x+1))`
`rArr" "A(x^(2)-x+1)+(Bx+C)(x+1)=1`
समान घातों के गुणांकों की तुलना करने पर
`A+B=0`
`-A+B+C=0`
`A+C=1`
हल करने पर
`A=(1)/(3),B=-(1)/(3),C=(2)/(3)`
`therefore int(1)/(1+x^(3))dx`
`=(1)/(3)int(1)/(1+x^(3))dx`
`=(1)/(3)int(1)/(x+1)dx+int(-(1)/(3)x+(2)/(3))/(x^(2)-x+1)dx`
`=(1)/(3)log|x+1|-(1)/(3)int(x-2)/(x^(2)-x+1)`
`=(1)/(3)log|x+1|-(1)/(6)int(2x-4)/(x^(2)-x+1)dx`
`=(1)/(3)log|x+1|-(1)/(6)int((2x-1)-3)/(x(2)-x+1)dx`
`=(1)/(3)log|x+1|-(1)/(6)int(2x-1)/(x^(2)-x+1)dx+(1)/(2)int(1)/(x^(2)-x+1)dx`
`=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(2)int(1)/((x-(1)/(2))^(2)+((sqrt3)/(2))^(2))dx`
`=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(2).(1)/(sqrt3//2)tan^(-1).(x-(1)/(2))/((sqrt3)/(2))+c`
`=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(sqrt3)tan^(-1).(2x-1)/(sqrt3)+c`
Discussion
409.

`int(x^(2))/(x^(2)+4)dx` का मान है -A. `x-4 tan^(-1)((x)/(2))+c`B. `x+4 tan^(-1)((x)/(2))+c`C. `x-2 tan^(-1)((x)/(2))+c`D. इनमे से कोई नहीं

Answer» Correct Answer - C
Discussion
410.

`intsqrtxdx` का मान ज्ञात कीजिए।

Answer» `intsqrtxdx=intx^(1//2)dx`
`=(x^((1)/(2)+1))/((1)/(2)+1)+c=(2)/(3).x^(3//2)+c`
Discussion
411.

`int(1)/(x^(2))dx` का मान ज्ञात कीजिए।

Answer» `int(1)/(x^(2))dx=intx^(-2)dx=(x^(-2+1))/(-2+1)+c=-(1)/(x)+c`
Discussion
412.

फलन का x के सापेक्ष समाकलन कीजिए -`x^(5//2)`

Answer» Correct Answer - `(2)/(7)x^(7//2)`
Discussion
413.

फलन का x के सापेक्ष समाकलन कीजिए -`x^(7)`

Answer» Correct Answer - `(x^(8))/(8)`
Discussion
414.

`int_(-2)^(2)x sin^(10)xdx` का मान है -A. 2B. `-2`C. 4D. इनमे से कोई नहीं

Answer» Correct Answer - D
Discussion
415.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/(3+2 sinx+cosx)dx`

Answer» `tan^(-1)(1+tan.(x)/(2))+c`
Discussion
416.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/(2+cosx)dx`

Answer» `(2)/(sqrt3)tan^(-1)((tan.(x)/(2))/(sqrt3))+c`
Discussion
417.

निम्नलिखित के मान ज्ञात कीजिए - `int(x^(2)+1)/(x^(4)-2x^(2)+1)dx`

Answer» Correct Answer - `(-x)/(x^(2)-1)+c`
Discussion
418.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/(3+4sinx)dx`

Answer» `(1)/(sqrt7 )log|(3 tan.(x)/(2)+4-sqrt7)/(3 tan.(x)/(2)+4+sqrt7)|+c`
Discussion
419.

`int_(0)^(pi//2)x sin x cosx dx` का मान है -A. `(pi)/(3)`B. `(pi)/(4)`C. `(pi)/(8)`D. `(pi)/(12)`

Answer» Correct Answer - C
Discussion
420.

`int_(0)^(1)(x)/(sqrt(1+x^(2)))dx` का मान है -A. `sqrt2+1`B. `sqrt2-1`C. `sqrt2`D. `-sqrt2`

Answer» Correct Answer - B
Discussion
421.

`int_(0)^(1)sqrt((1-x)/(1+x))dx` का मान है -A. `(pi)/(2)`B. `(pi)/(2)+1`C. `(pi)/(2)-1`D. इनमे से कोई नहीं

Answer» Correct Answer - C
Discussion
422.

`int_(0)^(1)x(1-x)^(n)dx` का मान है -A. `(1)/(n(n+1))`B. `(1)/((n+1)(n+2))`C. `(1)/(n(n+2))`D. `(1)/((n+1)(n+3))`

Answer» Correct Answer - B
Discussion
423.

`int_(2)^(8)|x-5|dx` का मान है -A. 8B. 9C. 10D. 11

Answer» Correct Answer - B
Discussion
424.

`int_(-2)^(2)log((7-x)/(7+x))dx` का मान है -A. 4B. `-4`C. 0D. 1

Answer» Correct Answer - C
Discussion
425.

फलन का x के सापेक्ष समाकलन कीजिए -`sec^(2)x-tan^(2)x`

Answer» Correct Answer - `x`
Discussion
426.

निम्नलिखित के मान ज्ञात कीजिए - `int(x^(2)+1)/(x^(4)+1)dx`

Answer» `(1)/(sqrt2)tan^(-1)((x^(2)-1)/(xsqrt2))+c`
Discussion
427.

फलन का x के सापेक्ष समाकलन कीजिए -`(1)/(1-cosx)`

Answer» Correct Answer - `-(cotx-cosec x)`
Discussion
428.

`int(1)/(sin^(2)x cos^(2)x)dx` का मान है -A. `tanx+cotx+c`B. `tanx-cotx+c`C. `secx+"cosec"x+c`D. `secx-"cosec"x+c`

Answer» Correct Answer - B
Discussion
429.

फलन का x के सापेक्ष समाकलन कीजिए -`cot^(2)x`

Answer» Correct Answer - `-cotx -x `
Discussion
430.

निम्नलिखित के मान ज्ञात कीजिए - `int(x^(2)-1)/(x^(4)+x^(2)+1)dx`

Answer» `(1)/(2)log|(x^(2)-x+1)/(x^(2)+x+1)|+c`
Discussion
431.

फलन का x के सापेक्ष समाकलन कीजिए -`(cos2x)/(cos^(2)x sin^(2)x)`

Answer» Correct Answer - `-(cot x -tanx )`
`int (cos2x)/(cos^(2)x sin^(2)x)dx=int (cos^(2)x-sin^(2)x)/(cos^(2)xsin^(2)x)dx`
`=int cosec^(2)x dx - int sec^(2)x dx `
Discussion
432.

फलन का x के सापेक्ष समाकलन कीजिए -`(sin^(2)x -cos^(2)x)/(sin^(2)xcos^(2)x)`

Answer» Correct Answer - `tanx +cotx `
`int (sin^(2)x-cos^(2)x)/(sin^(2)xcos^(2)x)dx=int (sin^(2)x)/(sin^(2)x cos^(2)x)dx-int (cos^(2)x)/(sin^(2)xcos^(2)x)dx`
`=int(1)/(cos^(2)x)dx-int (1)/(sin^(2)x)dx`
`=int sec^(2)xdx -int cosec^(2)x dx`
`=tanx + cotx +c`
Discussion
433.

फलन का x के सापेक्ष समाकलन कीजिए - `((4-7sinx)/(cos^(2)x))`

Answer» Correct Answer - `4tanx-7secx `
Discussion
434.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/(1+3 cos^(2)x)dx`

Answer» `(1)/(2)tan^(-1)((tanx)/(2))+c`
Discussion
435.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/((2sinx+cosx))dx`

Answer» `(-1)/(2(2tanx+1))+c`
Discussion
436.

`int(e^(2x)-1)/(e^(2x)+1)dx` का मान है -A. `log(e^(2x)+1)+c`B. `log(1+e^(-2x))+c`C. `log(e^(x)-e^(-x))+c`D. `log(e^(x)+e^(-x))+c.`

Answer» Correct Answer - D
Discussion
437.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/(2+sin^(2)x)dx`

Answer» `(1)/(sqrt6)tan^(-1)((sqrt3 tanx)/(sqrt2))+c`
Discussion
438.

निम्नलिखित के मान ज्ञात कीजिए- `int(1)/((a sinx+b cosx)^(2))dx`

Answer» `(-1)/(a(atanx+b))+c`
Discussion
439.

`(4x+1)/(sqrt(2x^(2)+x-3))`

Answer» `int(4x+1)/(sqrt(2x^(2)+x-3))" "{:("माना "2x^(2)+x-3=t^(2)),(rArr (4x+1)dx=2t dt):}`
`=int(2tdt)/(sqrt(t^(2)))=2 int 1 dt = 2t+c`
`=2sqrt(2x^(2)+x-3)+c`
Discussion
440.

`(1)/(sqrt((x-1)(x-b)))`

Answer» `int(1)/(sqrt((x-1)(x-b)))dx`
`=int(1)/(sqrt(x^(2)-x(a+b)+ab))dx`
`=int(1)/(sqrt((x-(a+b)/(2))^(2)+ab-((a+b)^(2))/(4)))dx`
`=int(1)/(sqrt((x-(a+b)/(2))^(2)-((a-b)/(2))^(2)))dx`
`=log|x-(a+b)/(2)+sqrt((x-(a+b)/(2))^(2)-((a-b)/(2))^(2))|+c`
`=log|x-(a+b)/(2)+sqrt((x-a)(x-b))|+c`
Discussion
441.

x के सापेक्ष समाकलन कीजिए `:` `(1)/(sqrt(8+3x-x^(2)))`

Answer» `int(1)/(sqrt(8+3x-x^(2)))dx`
`=int(1)/(sqrt(8-(x^(2)-3x+(9)/(4))+(9)/(4)))dx`
`=int(1)/(sqrt((sqrt(42))/(2))^(2)-(x-(3)/(2))^(2))dx`
`=sin^(-1)(x-(3)/(2))/(sqrt(41)//2)+c`
`=sin^(-1).(2x-3)/(sqrt(41))+c`
Discussion
442.

फलन का x के सापेक्ष समाकलन कीजिए - ` (1)/( sqrt ((x-1) (x -2)))`

Answer» `int(1)/(sqrt((x-1)(x-2)))dx`
`=int(1)/(sqrt(x^(2)-3x+2))dx`
`=int(1)/(sqrt((x^(2)-3x+(9)/(4))+2-(9)/(4)))dx`
`=int(1)/(sqrt((x-(3)/(2))^(2)-((1)/(2))^(2)))dx`
`=log|x-(3)/(2)+sqrt((x-(3)/(2))^(2)-((1)/(2))^(2))|+c`
`=log|x-(3)/(2)+sqrt(x^(2)-3x+2)|+c`
Discussion
443.

योगफल की सीमा के रूप में निम्नलिखित समाकलनों के मान ज्ञात कीजिए - `int_(-1)^(2)e^(-x)dx`

Answer» Correct Answer - `e-e^(-2)`
Discussion
444.

`sqrt(ax+b)`

Answer» `int sqrt(ax+b)dx" माना "ax+b=t`
`=intsqrtt (dt)/(a)=(1)/(a).intt^(1//2)dt" "rArr dx=dt`
`=(1)/(a).(t^(3//2))/(3//2)+c" "rArr dx=(dt)/(a)`
`=(2)/(3a)(ax+b)^(3//2)+c`
Discussion
445.

निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(0)^(2)(1)/(4+x-x^(2))dx`

Answer» `(1)/(sqrt(17)log)|(21+5sqrt(17))/(4)|`
Discussion
446.

निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(1)^(2)(1)/(x(1+x))dx`

Answer» Correct Answer - `log.(4)/(3)`
Discussion
447.

`int(2-3sinx)/(cos^(2)x)dx`

Answer» `int(2-3 sinx)/(cos^(2)x)dx=int((2)/(cos^(2)x)-(3sinx)/(cos^(2)x))dx`
`=2intsec^(2)x dx - 3 int tan x sec x dx`
`= 2 tan x-3 sec x+c`
Discussion
448.

निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(1)^(2)(x+3)/(x(x+2))dx`

Answer» Correct Answer - `(1)/(2)log6`
Discussion
449.

`(sqrtx+(1)/(sqrtx))` का प्रतिअवकलज है:A. `(1)/(3)x^((1)/(3))+2x^((1)/(2))+C`B. `(2)/(3)x^((2)/(3))+(1)/(2)x^(2)+C`C. `(2)/(3)x^((3)/(2))+2x^((1)/(2))+C`D. `(3)/(2)x^((3)/(2))+(1)/(2)x^((1)/(2))+C`

Answer» `int(sqrtx+(1)/(sqrtx))dx=intx^(1//2)dx+intx^(-1//2)dx`
`=(x^(3//2))/(3//2)+(x^(1//2))/(1//2)+c=(2)/(3)x^(3//2)+2x^(1//2)+c`
Discussion
450.

निम्नलिखित समाकलों के मान ज्ञात कीजिए - `int_(0)^(pi//2)x^(2)cosx dx`

Answer» Correct Answer - `((pi^(2))/(4)-2)`
Discussion