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Show that : int0^1(logx)/((1+x))dx=-int0...

Show that : `int_0^1(logx)/((1+x))dx=-int_0^1(log(1+x))/x dx`

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Verified by Experts

The correct Answer is:
NA
Let `I=int_(0)^(1)(logx)/((1+x))dx`
`=[log x log (1+x)]_(0)^(1)-int_(0)^(1)(log(1+x))/x dx`
`=0-int_(0)^(1)(log(1+x))/x dx`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
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  2. Prove that int0^(102)(x-1)(x-2)..(x-100)xx(1/(x-1)+1/(x-2)+.... .+ 1/...

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  3. Show that : int0^1(logx)/((1+x))dx=-int0^1(log(1+x))/x dx

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  4. If int0^1(e^t)/(1+t)dt=a , then find the value of int0^1(e^t)/((1+t)^2...

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  5. Let f be a one-to-one continuous function such that f(2)=3 and f(5)=7....

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  6. Evaluate: ("lim")(n rarr oo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))+ .......

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  7. Evaluate: underset(nrarrinfty)lim[(1/n^2sec^2(1/(n^2))+2/n^2sec^2((4)/...

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  8. Evaluate ("lim")(n rarr oo)sum(k=1)^nk/(n^2+k^2)

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  9. Evaluate the following limit: lim(nto oo)(sum(r=1)^(n) sqrt(r)sum(r=...

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  10. Evaluate the following limit: lim(nto oo)[(n!)/(n^(n))]^(1//n)

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  11. Prove that 4 lt= int1^3 sqrt(3+x^2) dx lt= 4sqrt(3)

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  12. If I1=int0^1 2^(x^2)dx , I2=int0^1 2^(x^3) dx , I3= int1^2 2^(x^2) dx ...

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  13. If II=int0^(pi//2)cos(sinx)dx ,I2=int0^(pi/2)sin(cosx)dx \ ,a n d \ I3...

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  14. Show that : pi/6leint0^1dx/sqrt(4-x^2-x^3)lepi/(4sqrt2)

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  15. Evaluate int(0)^(pi//2)|sinx-cosx|dx.

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  16. Evaluate: int(-1)^4f(x)dx=4a n dint2^4(3-f(x))dx=7, then find the val...

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  17. Evaluate int(1)^(5)sqrt((x-2)sqrt(x-1))dx.

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  18. Evaluate: int(-1)^3(tan^(-1)x/(x^2+1)+tan^(-1)(x^2+1)/x)dx

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  19. Evaluate int(1)^(a)x.a^(-[log(a)x])dx,(agt1).Here [.] represents the g...

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  20. Evaluate: int1^(e^6)[(logx)/3]dx ,w h e r e[dot] denotes the greatest...

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