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If I=int(e^x)/(e^(4x)+e^(2x)+1) dx. J=in...

If `I=int(e^x)/(e^(4x)+e^(2x)+1) dx. J=int(e^(-x))/(e^(-4x)+e^(-2x)+1) dx.` Then for an arbitrary constant c, the value of `J-I` equal to

A

`(1)/(2)log((e^(4x)-e^(2x)+1)/(e^(4x)+e^(2x)+1))+C`

B

`(1)/(2)log((e^(2x)+e^(x)+1)/(e^(2x)-e^(x)+1))+C`

C

`(1)/(2)log((e^(2x)+e^(x)+1)/(e^(2x)+e^(x)+1))+C`

D

`(1)/(2)log((e^(2x)+e^(2x)+1)/(e^(2x)+e^(2x)+1))+C`

Text Solution

Verified by Experts

The correct Answer is:
c
We have , `J-I=int(e^(3x))/(e^(4x)+e^(2x)+1)-(e^(x))/(e^(4)+e^(2x)+1)dx`
`rArrJ-I=int(e^(2x)-1)/(e^(4x)+e^(2x)+1)d(e^(x))`
`rArrJ-I=int(t^(2)-1)/(t^(4)+t^(2)+1)dt` where `t=e^(x)`
`rArrJ-I=int(1)/((t+(1)/(t))^(2)-1)d(t-(1)/(t))`
`rArrJ-I=(1)/(2)log((t+(1)/(t)-1)/(t+(1)/(t)+1))+C`
`rArrJ-I=(1)/(2)log((t^(2)-t+1)/(t^(2)+t+1))+C=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C`
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OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Chapter Test
  1. If I=int(e^x)/(e^(4x)+e^(2x)+1) dx. J=int(e^(-x))/(e^(-4x)+e^(-2x)+1) ...

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  2. The integral int (2x-3)/(x^2+x+1)^2 .dx is equal to

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  3. If int(xtan^(-1)x)/sqrt(1+x^(2))dx = sqrt(1+x^(2))f(x) + A " ln "sqrt(...

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  4. "If " int xlog(1+1//x)dx=f(x)log(x+1)+g(x)x^(2)+Ax+C, then

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  5. If int(xe^(x))/(sqrt(1+e^(x)))dx=f(x)sqrt(1+e^(x))-2logg(x)+C, then

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  6. The value of int (cos^3x+cos^5)/(sin^2x+sin^4x)dx

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  7. If int(dx)/((x^(2)+1)(x^(2)+4))=k tan^(-1) x + l tan^(-1) . (x)/(2) +C...

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  8. If int log(sqrt(1-x)+sqrt(1+x))dx=xf(x)+Ax+Bsin^(-1)x+C, then

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  9. If int(x^(5))/(sqrt(1+x^(3)))dx is equal to

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  10. The value of : inte^(secx).sec^(3)x(sin^(2)x+cosx+sinx+sinxcosx)dx i...

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  11. If int(2x^(2)+3)/((x^(2)-1)(x^(2)+4))dx=aln((x-1)/(x+1))+btan^(-1).(x)...

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  12. Let f(x)=(x)/((1+x^(n))^(1//n)) for n ge 2 and g(x)=underset("n times"...

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  13. The value of int ((ax^(2)-b)dx)/(xsqrt(c^(2)x^(2)-(ax^(2)+b)^(2))) is ...

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  14. Evalaute: inte^(x)(1+nx^(n-1)-x^(2n))/((1-x^(n))sqrt(1-x^(2n))dx

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  15. int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2)))dx

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  16. int(x^(3))/((1+x^(2))^(1//3))dx is equal to

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  17. int sinx/sin(x-alpha)dx=Ax+B log (sin(x-alpha))+C then find out (A ,B)

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  18. What is int (x^(2) +1)/(x^(4) - x^(2) + 1) dx equal to ?

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  19. Evaluate: int(x-1)/((x+1)sqrt(x^3+x^2+x))dx

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  20. int(1+x^(2))/(xsqrt(1+x^(4)))dx is equal to

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  21. int(1+x^(4))/((1-x^(4))^(3//2))dx is equal to

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