Question

How do you use Integration by Substitution to find `intx/(x^4+1)dx`?

Answer 1

Let `u=x^2`.
By taking the derivative,
`{du}/{dx}=2x`
by taking the reciprocal,
`Rightarrow{dx}/{du}=1/{2x}`
by multiplying by `du`,
`Rightarrow dx={du}/{2x}`

By rewriting the integral in terms of `u`,
`intx/{x^4+1}dx=intx/{u^2+1}cdot{du}/{2x}`
by cancelling out `x`'s,
`=1/2 int1/{1+u^2}du=1/2tan^{-1}u+C`
by putting `u=x^2` back in,
`=1/2tan^{-1}(x^2)+C`