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351.

`int ((x+2)(4x^(2)-5))/(x)dx` का मान ज्ञात कीजिए |

Answer» माना `I=int((x+2)(4x^(2)-5))/(x)dx`
`=int(4x^(3)+8x^(2)-5x-10)/(x)dx`
`=int((4x^(3))/(x)+(8x^(2))/(x)-(5x)/(x)-(10)/(x))dx`
`=4intx^(2)+8intxdx-5intdx-10int(dx)/(x)`
`=(4)/(3)x^(3)+4x^(2)-5x-10logx`
Discussion
352.

`int (secx)/(secx+tanx)dx` का मान ज्ञात कीजिए |

Answer» माना `I=int(secx)/(secx+tanx)dx`
अंश व हर को ` secx -tanx)` से गुणा करने पर
`I=int(secx(secx+tanx))/((secx+tanx)(secx-tanx))dx`
`=int(sec^(2)x-secxtanx)dx`
`(because sec^(2)x-tan^(2)x=1)`
`=tanx-secx`
Discussion
353.

`int(6sinx)dx` का मान ज्ञात कीजिए ।

Answer» `int(6sinx)dx=6intsinx dx`
`=6(-cosx)+c=-6cosx+c`
Discussion
354.

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

Answer» `int(x+(1)/(x))^(2)dx=int(x^(2)+2+(1))/(x^(2))dx`
`" "=intx^(2)dx+2int1dx+intx^(-2)dx`
`" "=(x^(3))/(3)+2x+(x^(-1))/(-1)+c`
`=(x^(3))/(3)+2x-(1)/(x)+c`
Discussion
355.

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

Answer» `(x^(2))/(2)+(2)/(3)x-(1)/(3)tan^(-1)x +c`
Discussion
356.

`int(3e^(x)-(1)/(5x)+secx tanx)dx` का मान ज्ञात कीजिए ।

Answer» `int(3e^(x)-(1)/(5x)+secx tanx)dx`
`=3inte^(x)dx-(1)/(5)int(1)/(x)dx+intsecx tanx dx`
`=3.e^(x)-(1)/(5)log_(e)x+secx+c`
Discussion
357.

`int(4x^(2)+x+1)/(x^(3)-1)dx` का मान है -A. `log(x^(3)-1)-log(x_(3)-1)+c`B. `log(x^(3)-1)-log(x^(3)-1)+c`C. `log(x^(3)-1)+log(x-1)+c`D. इनमे से कोई नहीं |

Answer» Correct Answer - C
Discussion
358.

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

Answer» Correct Answer - `x-tan^(-1)x+c`
Discussion
359.

मान ज्ञात कीजिए - `int (x^(3)+3x+4)/(sqrt(x))dx`

Answer» `(2)/(7)x^(7//2)+2x^(3//2)+8sqrt(x)+c`
Discussion
360.

यदि `(dy)/(dx)=4x^(3)-3e^(x)+5` तब y का मान ज्ञात कीजिए ।

Answer» Correct Answer - `x^(4)-3e^(x)+5x`
Discussion
361.

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

Answer» Correct Answer - `(x^(3))/(3)+x+c`
Discussion
362.

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

Answer» `2x^(1//2)+(4)/(3)x^(3//2)+(2)/(5)x^(5//2)+c`
Discussion
363.

यदि `(ds)/(dt)=2t^(2)+t-1` तथा s =0 जब t =0 , तब s तथा t में संबंध ज्ञात कीजिए ।

Answer» Correct Answer - `s=(2)/(3)t^(3)+(1)/(2)t^(2)-t`
`(ds)/(dt)=2t^(2)+t-1implies ds=(2t^(2)+t-1)dt`
`implies int ds =int (2t^(2)+t-1)dt`
`implies s=(2)/(3)t^(3) +(1)/(2)t^(2)-t`
t=0 पर s=0
Discussion
364.

मान ज्ञात कीजिए - `int (1-x)sqrt(x)dx`

Answer» `(2)/(3)x^(3//2)-(2)/(5)x^(5//2)+c`
Discussion
365.

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

Answer» Correct Answer - `(1)/(3)x^(3)-(1)/(x)-2x+c`
`int (x-(1)/(x))^(2)dx=int(x^(2)+(1)/(x^(2)-2)dx=(x^(3))/(3)-(1)/(x)-2x+c`
Discussion
366.

मान ज्ञात कीजिए - `int sin^(2)(x)/(2)dx`

Answer» Correct Answer - `(x)/(2)-(1)/(2)sinx +c`
`int sin^(2)(x)/(2)dx=int (1)/(2)(1-cosx)dx`
`=int ((1)/(2)-(1)/(2)cosx)dx=(x)/(2)-(1)/(2)sinx +c`
Discussion
367.

मान ज्ञात कीजिए - `int (2tanx -3cotx)^(2)dx`

Answer» Correct Answer - `4tanx-9cotx-25x+c`
Discussion
368.

मान ज्ञात कीजिए -`int sqrt(x)(3x^(2)+2x+3)dx`

Answer» `(6)/(7)x^(7//2)+(4)/(5)x^(5//2)+2x^(3//2)+c`
Discussion
369.

मान ज्ञात कीजिए -`int (2x-3cosx+e^(x))dx`

Answer» `x^(2)-3sinx+2^(x)+c`
Discussion
370.

मान ज्ञात कीजिए - `int (2+3cosx)/(sin^(2)x)dx`

Answer» Correct Answer - `-2cotx -3cosecx+c`
Discussion
371.

मान ज्ञात कीजिए -`int (2x-3cosx+e^(x))dx`

Answer» `x^(2)-3sinx+2^(x)+c`
Discussion
372.

`int(sinx)/((1+cosx)(2+3cosx))dx` का मान ज्ञात कीजिए ।

Answer» माना `I=int(sinx)/((1+cosx)(2+3cosx))dx`
`=int(-dt)/((1+t)(2+3t))`
`" माना "cosx=t`
`" "-sinx=(dt)/(dx)`
`" "rArr sinx dx =-dt`
माना `(-1)/((1+t)(2+3t))=(A)/(1+t)+(B)/(2+3t)=(A(2+3t)+B(1+t))/((1+t)(2+3t))`
`rArr" "A(2+t)+B(1+t)=-1`
`t=-1` रखने पर
`A(2-3)+0=-1`
`rArr" "A=1`
`t=-(2)/(3)` रखने पर
`0+B(1-(2)/(3))=-1`
`rArr" "B=-3`
`therefore" "I=int((1)/(1+t)-(3)/(2+3t))dt`
`=log|1+t|-log|2+3t|+c`
`=log|(1+t)/(2+3t)|+c`
`=log|(1+cosx)/(2+3cosx)|+c`
Discussion
373.

`int_(0)^(pi//4)tanx. secxdx` का मान ज्ञात कीजिए।

Answer» `int_(0)^(pi//4)tan x sec x dx`
`=[secx]_(0)^(pi//4)=sec.(pi)/(4)-sec0 = (sqrt2-1)`
Discussion
374.

`int(sinx)/(cos^(2)x)` का मान ज्ञात कीजिए ।

Answer» `int(sinx)/(cos^(2)x)dx=int tanx.secxdx=secx+c`
Discussion
375.

`int sqrt(tanx)dx` का मान ज्ञात कीजिए ।

Answer» माना
`I=intsqrt(tanx)dx`
`=int sqrt(t^(2)).(2tdt)/(1+t^(4))" "{:("माना "tanx=t^(2)),(therefore sec^(2)xdx=dt.2t):}`
`=int((t^(2)+1)+(t^(2)-1))/(t^(4)+1)dt" "rArr dx=(2t dt)/(1+tan^(2)x)`
`=int(t^(2)+1)/(t^(4)+1)dt+int(t^(2)-1)/(t^(4)+1)dt" "=(2tdt)/(1+t^(4))`
`=int(1+(1)/(t^(2)))/(t^(2)+(1)/(t^(2)))dt+int(1-(1)/(t^(2)))/(t^(2)+(1)/(t^(2)))dt`
`=int(1+(1)/(t^(2)))/((t-(1)/(t))^(2)+(sqrt2)^(2))dt+int(1-(1)/(t^(2)))/((t+(1)/(t))^(2)-(sqrt2)^(2))dt`
माना प्रथम समाकल के लिये `t-(1)/(t)=u rArr (1+(t)/(t^(2)))dt=du`
दूसरे समाकल के लिये `t+(1)/(t)= v rArr (1-(1)/(t^(2)))dt=dv`
`therefore" "I=int(1)/(u^(2)+(sqrt2)^(2))du+int(1)/(v^(2)-(sqrt2)^(2))dv`
`=(1)/(sqrt2)tan^(-1)((u)/(sqrt2))+(1)/(2sqrt2)log|(v-sqrt2)/(v+sqrt2)|+c`
Discussion
376.

`int(e^(loge))/(x)dx` का मान ज्ञात कीजिए ।

Answer» `int(e^(logx))/(x)dx=int(x)/(x)dx=int1dx=x+c`
Discussion
377.

`int(1)/(x^(4)+1)dx` का मान ज्ञात कीजिए ।

Answer» माना `I=int(1)/(x^(4)+1)dx=(1)/(2)int(2)/(x^(4)+1)dx`
`=(1)/(2)int((x^(2)+1)-(x^(2)-1))/(x^(4)+1)dx`
`=(1)/(2)int(x^(2)+1)/(x^(4)+1)dx-(1)/(2)int(x^(2)-1)/(x^(4)+1)dx`
`=(1)/(2)int(1+(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx-(1)/(2)int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx`
`rArr" "I=(1)/(2)int(1+(1)/(x^(2)))/((x-(1)/(x))^(2)+(sqrt2)^(2))dx-(1)/(2)int(1-(1)/(x^(2)))/((x+(1)/(x))^(2)-(sqrt2)^(2))dx`
माना प्रथम समाकल के लिये
`x-(1)/(x)=t rArr (1+(1)/(x^(2)))dx=dt`
और दूसरे समाकल के लिये
`x+(1)/(x)=u rArr (1-(1)/(x^(2)))dx=du`
`therefore I=(1)/(2)int(dt)/(t^(2)+(sqrt2)^(2))-(1)/(2)int(du)/(u^(2)-(sqrt2)^(2))`
`=(1)/(2sqrt2)tan^(-1).(t)/(sqrt2)-(1)/(2).(1)/(2sqrt2)log|(u-sqrt2)/(u+sqrt2)|+c`
`=(1)/(2sqrt2)tan^(-1)((x-(1)/(x))/(sqrt2))-(1)/(4sqrt2)log|(x+(1)/(x)-sqrt2)/(x+(1)/(x)+sqrt2)|+c`
`=(1)/(2sqrt2)tan^(-1)((x^(2)-1)/(xsqrt2))-(1)/(4sqrt2)log|(x^(2)-xsqrt2+1)/(x^(2)+xsqrt2+1)|+c`
Discussion
378.

`int(x^(2)+1)/(x^(4)+x^(2)+1)dx` का मान ज्ञात कीजिए ।

Answer» `int(x^(2)+1)/(x^(4)+x^(2)+1)dx=int(1+(1)/(x^(2)))/(x^(2)+1+(1)/(x^(2)))dx`
(अंश और हर को `x^(2)` से भाग देने पर )
`=int(1+(1)/(x^(2)))/((x-(1)/(x))^(2)+(sqrt3)^(2))dx" "{:("माना "x-(1)/(x)=t),(rArr(1+(1)/(x^(2)))dx=dt):}`
`=int(1)/(t^(2)+(sqrt3)^(2))dt=(1)/(sqrt3)tan^(-1)((t)/(sqrt3))+c`
`=(1)/(sqrt3)tan^(-1)((x-(1)/(x))/(sqrt3))+c`
Discussion
379.

`int(1)/(4+5 sin x)dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(4+5sinx)dx`
`=int(1)/(4(sin^(2).(x)/(2)+cos^(2).(x)/(2))+5.2sin.(x)/(2)cos.(x)/(2))dx`
`=(1)/(2)int(sec^(2).(x)/(2))/(2 tan^(2).(x)/(2)+2+5tan.(x)/(2))dx`
(अंश और हर को `cos^(2).(x)/(2)` से भाग देने पर )
`int(dt)/(2t^(2)+5t+2)" माना "tan.(x)/(2)=t`
`=int(1)/((t+2)(2t+1))dt" "rArr (1)/(2)sec^(2).(x)/(2)dx=dt` ltBrgt `=(2)/(3) int(1)/(2t+1)dt -(1)/(3)int(1)/(t+2)dt`
(आंशिक भिन्न में बदलने पर )
`=(1)/(3)log|2t+1|-(1)/(3)log|t+2|+c`
`=(1)/(3)log|(2t+1)/(t+2)|+c`
`=(1)/(3)log|(2tan.(x)/(2)+1)/(tan.(x)/(2)+2)|+c`
Discussion
380.

`int(4+3sinx)/(cos^(2)x)dx` को हल कीजिए ।

Answer» `int(4+3sinx)/(cos^(2)x)dx`
`=int((4)/(cos^(2)x)+(3sinx)/(cos^(2)x))dx`
`=4intsec^(2)xdx+3intx.secxdx`
`=4.tanx+3secx+c`
Discussion
381.

`int(1)/(5+4 cosx)dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(5+4 cosx)dx`
`=int(1)/(5(cos^(2).(x)/(2)+sin^(2).(x)/(2))+4(cos^(2).(x)/(2)-sin^(2).(x)/(2)))dx`
`=int(1)/(9cos^(2).(x)/(2)+sin^(2).(x)/(2))dx`
(अंश और हर को `cos^(2).(x)/(2)` से भाग देने पर )
`=int(sec^(2).(x)/(2)dx)/(9+tan^(2).(x)/(2))`
`=int(2dt)/(9+t^(2))" माना " tan.(x)/(2)=t`
`=2 int(1)/(t^(2)+3^(2))dt" "therefore (1)/(2)sec^(2).(x)/(2)=(dt)/(dx)`
`=(2)/(3) tan^(-1)((t)/(3))+c" "rArr sec^(2).(x)/(2)dx=2dt`
`=(2)/(3)tan^(-1)((tan.(x)/(2))/(3))+c`
Discussion
382.

`int(secx+tanx)/(secx-tanx)dx` को हल कीजिए ।

Answer» `int(secx+tanx)/(secx-tanx)dx`
`=int((secx+tanx)^(2))/(sec^(2)x-tan^(2)x)dx`
`=int(sec^(2)x+tan^(2)x+2secxtanx)dx`
`=int{sec^(2)x+(sec^(2)x-1)+2secxtanx}dx`
`=2intsec^(2)xdx - int 1dx+2intsecxtanxdx`
`=2tanx-x+2secx+c`
Discussion
383.

`intsec^(2)3xdx` का मान ज्ञात कीजिए ।

Answer» `intsec^(2)3xdx=(1)/(3)tan3x+c`
Discussion
384.

`int(1)/(a^(2)cos^(2)x+b^(2)sin^(2)x)dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(a^(2)cos^(2)x+b^(2)sin^(2)x)dx`
`=int(sec^(2)x)/(a^(2)+b^(2)tan^(2)x)dx`
(अंश और हर को `cos^(2)x` से भाग देने पर )
`=int(dt)/(b(a^(2)+t^(2)))" "{:("माना "b tan x=t),(therefore" "b sec^(2)x=(dt)/(dx)):}`
`=(1)/(b).(1)/(a)tan^(-1)((t)/(a))+c rArr sec^(2)x dx =(dt)/(b)`
`=(1)/(ab)tan^(-1)((b tanx)/(a))+c`
Discussion
385.

`intcosxsqrt(4-sin^(2)x)dx` का मान ज्ञात कीजिए ।

Answer» `intcosxsqrt(4-sin^(2)x)dx" माना "sinx=t`
`=intsqrt(4-t^(2))dt" "therefore cos xdx = dt`
`=intsqrt(2^(2)-t^(2))dt`
`=(t)/(2)sqrt(2^(2)-t^(2))+(2^(2))/(2)sin^(-1)((t)/(2))+c`
`=(sinx)/(2)sqrt(4-sin^(2)x)+2 sin^(-1)((sinx)/(2))+c`
Discussion
386.

`intsqrt(4-5x)dx` का मान ज्ञात कीजिए ।

Answer» `intsqrt(4-5x)dx=int(4-5x)^(1//2)dx`
`" "=(1)/(-5).((4-5x)^(3//2))/(3//2)+c`
`" "=-(2)/(15)(4-5x)^(3//2)+c`
Discussion
387.

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

Answer» `int(1)/(1+cos^(2)x)dx`
`=int(1)/((sin^(2)x+cos^(2)x)+cos^(2)x)dx`
`=int(1)/(sin^(2)x+2cos^(2)x)dx`
`=int(sec^(2)x)/(tan^(2)x+2)dx`
(अंश और हर को `cos^(2)x` से भाग देने पर )
`=int(1)/(t^(2)+(sqrt2)^(2))dt" माना "tax=t`
`=(1)/(sqrt2)tan^(-1)((t)/(sqrt2))+c" " rArrsec^(2)xdx=dt`
`=(1)/(sqrt2)tan^(-1)((tanx)/(sqrt2))+c`
Discussion
388.

`intsqrt(4x^(2)+9)dx` का मान ज्ञात कीजिए ।

Answer» `int sqrt(4x^(2)+9)dx" माना "2x=t`
`=intsqrt((2x)^(2)+3^(2))dx" "thereofore 2=(dt)/(dx)`
`=intsqrt(t^(2)+3^(2)).(dt)/(2)" "rArr" "dx=(dt)/(2)`
`=(1)/(2)[(t)/(2)sqrt(t^(2)+3^(2))+(3^(2))/(2)log|t+sqrt(t^(2)+3^(2))|]+c`
`=(1)/(4)[2xsqrt(4x^(2)+9)+9log |2x+sqrt(4x^(2)+9)|]+c`
Discussion
389.

`intsqrt(2x-5x-1)dx` का मान ज्ञात कीजिए ।

Answer» `intsqrt(2x^(2)-5x-1)dx`
`=sqrt2 int sqrt(x^(2)-(5)/(2)x-(1)/(2))dx`
`=sqrt2 intsqrt((x^(2)-(5)/(2)x+(25)/(16))-((1)/(2)+(25)/(16)))dx`
`=sqrt2intsqrt((x-(5)/(4))^(2)-((sqrt(33))/(4))^(2))dx`
`=sqrt2[(1)/(2)(x-(5)/(4))sqrt((x-(5)/(4))^(2)-((sqrt(33))/(4))^(2))-((sqrt(33)//4)^(2))/(2)log|x-(5)/(4)+sqrt((x-(5)/(4))^(2)-((sqrt(33))/(4))^(2))|]+c`
`=(1)/(sqrt2)(x-(5)/(4))sqrt(x^(2)-(5)/(2)x-(1)/(2))-(33)/(16sqrt2)log|x-(5)/(4)+sqrt(x^(2)-(5x)/(2)-(1)/(2))|+c`
`=(1)/(2)(x-(5)/(4))sqrt((2x^(2)-5x-1))-(33)/(16sqrt2)log|x-(5)/(4)+(1)/(sqrt2)sqrt(2x^(2)-5x-1)|+c`
Discussion
390.

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

Answer» `int(x^(2))/((x^(2)-1)(x^(2)+2))dx`
अब `(x^(2))/((x^(2)-1)(x^(2)+2))=(t)/((t-1)(t+2))`
`" "`(यहाँ `x^(2)=t` लेने पर )
`=(A)/(t-1)+(B)/(t+2)=(A(t+2)+B(t-1))/((t-1)(t+2))` (माना )
`rArr" "A(t+2)+B(t-1)=t`
t के गुणांकों की और अचर पदों की तुलना करने पर
`A+B=1`
`2A-B=0`
हल करने पर `A=(1)/(3),B=(2)/(3)`
`therefore" "(t)/((t-1)(t+2))=(1)/(3(t-1))+(2)/(3(t+2))`
`rArr" "(x^(2))/((x^(2)-1)(x^(2)+2))=(1)/(3(x^(2)-1))+(2)/(3(x^(2)+2))`
`rArr" "int(x^(2))/((x^(2)-1)(x^(2)+2))dx`
`" "=(1)/(3)int(1)/(x^(2)-1)dx+(2)/(3)int(1)/(x^(2)+2)dx`
`" "=(1)/(3).(1)/(2)log|(x-1)/(x+1)|+(2)/(3).(1)/.(sqrt2)tan^(-1).(x)/(sqrt2)+c`
`" "=(1)/(6)log|(x-1)/(x+1)|+(sqrt2)/(3)tan^(-1).(x)/(sqrt2)+c`
Discussion
391.

`int(x^(2)+x+3)/((x-2)(x+1))dx` का मान ज्ञात कीजिए ।

Answer» माना `I=int(x^(2)+x+3)/((x-2)(x+1))dx`
`=int(x^(2)+x+3)/(x^(2)-x-2)dx`
`=int(1+(2x+5)/(x^(2)-x-2))dx`
(अंश और हर की घात समान हैं अतः भाग देने पर )
`rArr" "I=int[1+(2x+5)/((x-2)(x+1))]dx`
माना `(2x+5)/((x-2)(x+1))=(A)/(x-2)+(B)/(x+1)=(A(x+1)+B(x-2))/((x-2)(x+1))`
`rArr" "A(x+1)+B(x-2)=2x+5`
x = 2 रखने पर
`A(2+1)+0=2(2)+5`
`rArr" "A=3`
`x=-1` रखने पर
`0+B(-1-2)=2(-1)+5`
`rArr" "B=-1`
`I=int(1+(3)/(x-2)-(1)/(x+1))dx`
`=x+3log(x-2)-log(x+1)+c`
Discussion
392.

`int(2x+1)/(sqrt(2x^(2)+x-3))dx` का मान ज्ञात कीजिए ।

Answer» माना `I=int(2x+1)/(sqrt(2x^(2)+x-3))dx`
माना `2x+1=A.(d)/(dx)(2x^(2)+x-3)+B`
`=A(4x+1)+B`
x के गुणांकों की और अचर पदों की तुलना करने पर
`{:(,2=4A," और ",1=A+B),(rArr.,A=1//2," और ",B=1//2):}`
अब `I=(1)/(2)int(4x+1)/(sqrt(2x^(2)+x-3))dx+(1)/(2)int(1)/(sqrt(2x^(2)+x-3))dx`
`=I_(1)+I_(2)` (माना )
`{:(I_(1),(1)/(2)int(4x+1)/(sqrt(2x^(2)+x-3))dx," माना ",2x^(2)+x-3=t),(,=(1)/(2)int(1)/(sqrtt).dt," "rArr.,4x+1=(dt)/(dx)),(,=sqrtyt," "rArr.,(4x+1)dx=dt),(,=sqrt(2x^(2)+x-3),,):}`
और `I_(2)=(1)/(2)int(1)/(sqrt(2x^(2)+x-3))dx`
`=(1)/(2sqrt2)int(1)/(sqrt(x^(2)+(1)/(2)x-(3)/(2)))dx`
`=(1)/(2sqrt2)int(1)/(sqrt((x^(2)+(1)/(2)x+(1)/(16))-((3)/(2)+(1)/(16))))dx`
`=(1)/(2sqrt2)int(1)/(sqrt((x+(1)/(4))^(2)-((5)/(4))^(2)))dx`
`=(1)/(2sqrt2)log|x+(1)/(4)+sqrt((x+(1)/(4))^(2)-((5)/(4))^(2))|`
`=(1)/(2sqrt2)log|x+(1)/(4)+sqrt(x^(2)+(1)/(2)x-(3)/(2))|`
`=(1)/(2sqrt2)log|(4x+1+sqrt(16x^(2)+8x-24))/(4)|`
`=(1)/(2sqrt2)log|4x+1+sqrt(16x^(2)+8x-24)|-(1)/(2sqrt2)log4`
`therefore I=sqrt(2x^(2)+x-3)+(1)/(2sqrt2)log|4x+1+sqrt(16x^(2)+8x-24)|+c`
`(because -(1)/(2sqrt2)log4 "अचर है ")`
Discussion
393.

`int secx tanx sqrt(4 sec^(2)x-1)dx` का मान ज्ञात कीजिए ।

Answer» `int sec x tan x sqrt(4 sec^(2)x-1)dx`
`=int sec x tanx sqrt((2secx)^(2)-1^(2))dx`
`=intsqrt(t^(2)-1^(2))(dt)/(2)" "{:(" माना "2sec x=t),(therefore" "2 sec x tanx=(dt)/(dx)),(therefore" "sec x tanx dx =(dt)/(2)):}`
`=(1)/(2)[(t)/(2)sqrt(t^(2)-1)-(1)/(2)log|t+sqrt(t^(2)-1)|]+c`
`=(1)/(4)[2 secxsqrt(4 sec^(2)x-1)-log|2secx+sqrt(4sec^(2)x-1)|]+c`
Discussion
394.

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

Answer» माना `I=int(2x)/((x^(2)+1)(x^(2)+2))dx`
`=int(1)/((t+1)(t+2))dt" माना "x^(2)=t`
`" "rArr 2x dx = dt`
माना `(1)/((t+1)(t+2))=(A)/(t+1)+(B)/(t+2)`
`=(A(t+2)+B(t+1))/((t+1)(t+2))`
`rArr" "A(t+2)+B(t+1)=1`
`t=-1` रखने पर
`A(-1+2)+0=1`
`rArr" "A=1`
`t=-2` रखने पर
`0+B(-2+1)=1`
`rArr" "B=-1`
`therefore" "I=int(1)/(t+1)dt-int(1)/(t+2)dt`
`=log|t+1|-log|t+2|+c`
`=log|(t+1)/(t+2)|+c=log|(x^(2)+1)/(x^(2)+2)|+c`
Discussion
395.

`int(1)/(4-x^(2))dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(4-x^(2))dx`
`=int(1)/(2^(2)-x^(2))dx=(1)/(2(2))log|(2+x)/(2-x)|+c`
`=(1)/(4)log|(2+x)/(2-x)|+c`
Discussion
396.

`int(1)/(sqrt(x^(2)-2x+10))dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(sqrt(x^(2)-2x+10))dx=int(1)/(sqrt((x^(2)-2x+1)+9))`
`=int(1)/(sqrt((x-1)^(2)+3^(2)))`
`log|x-1+sqrt((x-1)^(2)+3^(2))|+c`
`=log|x-1+sqrt(x^(2)-2x+10)|+c`
Discussion
397.

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

Answer» माना `(x)/((x-2)(x-1)^(2))`
`=(A)/(x-2)+(B)/(x-1)+(C)/((x-1)^(2))`
`=(A(x-1)^(2)+B(x-1)(x-2)+C(x-2))/((x-2)(x-1)^(2))`
`rArr A(x-1)^(2)+B(x-1)(x-2)+C(x-2)=x`
समान घातों के गुणांकों की तुलना करने पर
`A+B=0`
`-2A-3B+C=1`
`A+2B-2C=0`
हल करने पर `A=2, B=-2,C=1`
`therefore int(x)/((x-2)(x-1)^(2))dx`
`=int(2)/(x-2)dx- int(2)/(x-1)dx - int(1)/((x-1)^(2))dx`
`=2log|x-2|-2log|x-1|+(1)/(x-1)+c`
`=2log|(x-2)/(x-1)|+(1)/(x-1)+c`
Discussion
398.

`int(3x+1)/(2x^(2)-2x+3)dx` का मान ज्ञात कीजिए ।

Answer» माना `3x+1=A.(d)/(dx)(2x^(2)-2x+3)+B`
`rArr" "3x+1=A(4x-2)+B`
x के गुणांकों की तथा अचर पदों की तुलना करने पर
`{:(,3=4A," और ",1=-2A+B),(rArr.,A=(3)/(4)," और ",B=(5)/(2)):}`
`therefore int(3x+1)/(2x^(2)-2x+3)dx=int((3)/(4)(4x-2)+(5)/(2))/(2x^(2)-2x+3)dx`
`" "=(3)/(4)int(4x-2)/(2x^(2)-2x+3)dx+(5)/(2)int(1)/(2x^(2)-2x+3)dx`
`=(3)/(4)int(1)/(t)dt+(5)/(4)int(1)/(x^(2)-x+(3)/(2))dx`
माना `" "2x^(2)-2x+3=t`
`" "(4x-2)dx=dt`
`=(3)/(4)log|t|+(5)/(4)int(1)/((x^(2)-x+(1)/(4))+((3)/(2)-(1)/(4)))dx`
`=(3)/(4)log|2x^(2)-2x+3|+(5)/(4) int(1)/((x-(1)/(2))^(2)+((sqrt5)/(2))^(2))dx`
`=(3)/(4)log|2x^(2)-2x+3|+(5)/(4).(1)/(sqrt5//2)tan^(-1).(x-(1)/(2))/((sqrt5)/(2))+c`
`=(3)/(4)log|2x^(2)-2x+3|+(sqrt5)/(2)tan^(-1)((2x-1)/(sqrt5))+c`
Discussion
399.

`int(x^(2)+x+1)/((x-1)^(3))dx` का मान ज्ञात कीजिए ।

Answer» `int(x^(2)+x+1)/((x-1)^(3))dx`
`=int((t+1)^(2)+(t+1)+1)/(t^(3))`
`=int(t^(2)+3t+3)/(t^(3))" "{:("माना "x-1=t),(rArr" "dx=dt):}`
`=int((1)/(t)+(3)/(t^(2))+(3)/(t^(3)))dt=log|t|-(3)/(t)-(3)/(2)+c`
`=log|x-1|-(3)/(x-1)-(3)/(2(x-1)^(2))+c`
Discussion
400.

`int(1)/(4x^(2)+4x+5)dx` का मान ज्ञात कीजिए ।

Answer» `int(1)/(4x^(2)+4x+5)dx`
`=(1)/(4)int(1)/(x^(2)+x+(5)/(4))dx=(1)/(4)int(1)/((x^(2)+x+(1)/(4))+1)dx`
`=(1)/(4)int(1)/((x+(1)/(2))^(2)+1^(2))dx=(1)/(4).tan^(-1)((x+(1)/(2))/(1))+c`
`=(1)/(4)tan^(-1)((2x+1)/(2))+c`
Discussion