How do you find the integral of x^3/(1+x^2)?

1 Answer
Oct 10, 2015

1/2(1+x^2-ln(1+x^2)) + C

Explanation:

We'll do integration by substitution. Take:
u = x^2
du = 2x
Substituting this in, we get:

1/2\intu/(1+u)du
Now take:
t = 1 + u
dt = du

Substitute this in:
1/2\int(t-1)/tdt = 1/2\int1 - 1/tdt = 1/2(\int1dt - \int 1/tdt)
=1/2(t - ln(t)) + C
Now we need to change back to x:
=1/2(1+u-ln(1+u)) + C
=1/2(1+x^2-ln(1+x^2)) + C