Question

How do you find the integral of `(x^4+x-4) / (x^2+2)`?

Answer 1

First, use polynomial long division to get `(x^4+x-4)/(x^2+2)=x^2-2+x/(x^2+2)`. Next, integrate this expression, using a substitution `u=x^2+2, du=2x\ dx` to get:

`\int(x^4+x-4)/(x^2+2)dx=\int(x^2-2+x/(x^2+2))dx`

`=\frac{1}{3}x^3-2x+\frac{1}{2}ln(x^2+2)+C`

Note that absolute value signs are not needed in the argument of the logarithm function since `x^2+2>0` for all real `x`.