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int{(logx-1)/(1+(logx)^(2))}^(2) dx is e...

`int{(logx-1)/(1+(logx)^(2))}^(2)` dx is equal to

A

`(x)/((logx)^(2)+1)+C`

B

`(xe^(x))/(1+x^(2))+C`

C

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

D

`(logx)/((logx)^(2)+1)+C`

Text Solution

Verified by Experts

The correct Answer is:
A
We have , `I=int{(logx-1)/(1+(logx)^(2))}^(2)dx`
`rArr I=inte^(t)((t-1)^(1))/((t^(2)+1)^(2))dt` , where t = log x
`rArrI=inte^(t)(t^(2)+1-2t)/((t^(2)+1)^(2))dt`
`rArr I=inte^(t){(1)/(underset(f)(t^(2)+1))+(-2t)/(underset(f)((t^(2)+1)^(2)))}dt`
`rArrI=(e^(t))/(t^(2)+1)+C=(x)/((logx)^(2)+1)+C`
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OBJECTIVE RD SHARMA-INDEFINITE INTEGRALS-Chapter Test
  1. int{(logx-1)/(1+(logx)^(2))}^(2) dx is equal to

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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)+Aln|x+sqrt(x^2+1)...

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  4. Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C , then f(x)=1/2x^2 (b) ...

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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 the integral int (cos^3x+cos^5 x)/(sin^2 x+sin^4 x) dx is...

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  7. If int(1)/((x^(2)+1)(x^(2)+4))dx=Atan^(-1)x+B" tan"^(-1)(x)/(2)+C , t...

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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^3x(sin^2x+cosx+sinx+sinxcosx)dx is

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  11. int(2x^(2)+3)/((x^(2)-1)(x^(2)+4))dx=alog((x+1)/(x-1))+b"tan"^(-1)(x)/...

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  12. Let f(x)=x/((1+x^n)^(1/ n)) for ngeq2 and g(x)=(f(ofo ...of)(x) Then ...

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

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  14. Evaluate: inte^x(1+n x^(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^(4) -1)/(x^(2) sqrt(x^(4) + x^(2) + 1)) dx equal to ?

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

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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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