Find the integral of `x^3/(1+25x^2)` dx. I already know the answer but I cant find the solution step-by-step. Thank you in Advice.

Hello!

Express the function under integral as `(x^2*x)/(1+25x^2),` observe that `x*dx = 1/2 d(x^2)` and make the substitution `x^2=u.` Then `du=2xdx` and the integral becomes

`1/2 int (u)/(1+25u) du.`

It is simple to extract a proper part from the fraction `u/(1+25u),`

`u/(1+25u)=1/25 (1-1/(1+25u)).`

So the integral is

`1/50 int(1-1/(1+25u)) du = 1/50 (u-1/25 ln|1+25u|)+C,`

or in terms of x

`1/50 (x^2 - 1/25 ln(1+25x^2))+C.`