integration of log(1+x^3)

Dear Piyush

 I= ∫1* log(1+x^3) dx

intigrate by part  take 1 as second function

I= x log(1+x^3) -∫3x³/(1+x^3)  dx

I= x log(1+x^3) -3∫(x³ +1 -1 )/(1+x^3)  dx

I= x log(1+x^3) --3x +3∫1 /(1+x^3)  dx

now use partial fraction

1 /(1+x^3)  = 1/3(x+1)   + (2-x)/3(x²-x+1)

so intigral become

I= x log(1+x^3) --3x +∫1 /(1+x)  dx  + ∫2-x)/(x²-x+1)  dx

now it is simple intigration

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Badiuddin

its is frm of 1*log(1+x³) apply product rule ..take log as a 1 function and '1' as second function ....

and solve it........