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The value of int (1)^(e^(2)) (dx)/(x(1+...

The value of ` int _(1)^(e^(2)) (dx)/(x(1+ log x)^(2) ) ` is

A

`2/3`

B

`1/3`

C

`3/2`

D

log 2

Text Solution

Verified by Experts

The correct Answer is:
A
Let `l = int_(1)^(e^(2))(dx)/(x(1+log x)^(2))`
Put (1 + log x) `= t rArr dt = (1)/(x) dx`
When x = 1, t = 1
When `x=e^(2), t= 3`
`therefore " " l = int_(1)^(3)(dt)/(t^(2))=[(-1)/(t)]_(1)^(3)=-[(1)/(3)-1]=(2)/(3)`
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  4. int (0)^(3) (3x+1)/(x^(2)+9) dx =

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  5. underset(n to oo)lim underset(r=1)overset(n)sum(1)/(n)e^(r//n) is

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  7. The value of the integral int (0)^(pi//2) sin ^(5) x dx is

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  8. If int (0)^(x^(2)) f(t) dt = x cos pix , then the value of f(4) is

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  9. If f(x) is differentiable and int0^(t^2)xf(x)dx=2/5t^5, then f(4/(25))...

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  10. The value of int (0)^(pi//2) sin ^(8) x dx is

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  11. The value of overset(pi)underset(-pi)int(1-x^(2)) sin x cos^(2) x" dx"...

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  12. int (0)^(1) (xdx)/([x + sqrt(1-x^(2))]sqrt(1-x^(2))) is equal to

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  13. The value of int(0)^(pi)(Sigma(r=0)^(3)a(r)cos^(3-r)x sin^(r)x)dx depe...

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  14. The value of int (-1)^(1) x|x| dx is

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  15. The value of integral int (1//pi)^(2//pi)(sin(1/x))/(x^(2))dx=

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  16. The value of int (1)^(e^(2)) (dx)/(x(1+ log x)^(2) ) is

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  18. The value of the integral overset(1)underset(0)int x(1-x)^(n)dx, is

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  19. The value of int (0)^(pi) (dx)/(5+4 cos x ) is

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  20. The integral int (0)^(1) (dx)/(1-x+x^(2)) has the value

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