# Integrating (x² + 1)² using substitution

• Jul 29th 2012, 10:27 AM
SweatingBear
Integrating (x² + 1)² using substitution
Hey forum how would I go about in order to compute this indefinite integral,

$\int \, (x^2 +1)^2 \, \text{d}x \ ?$

I can't seem to eliminate all $x$-variables using $u = x^2 + 1$. I know I can simply compute it by expanding the integrand but I want to do it using substitution in particular.
• Jul 29th 2012, 10:42 AM
jeduhi
Re: Integrating (x² + 1)² using substitution
• Jul 29th 2012, 10:43 AM
Prove It
Re: Integrating (x² + 1)² using substitution
Quote:

Originally Posted by SweatingBear
Hey forum how would I go about in order to compute this indefinite integral,

$\int \, (x^2 +1)^2 \, \text{d}x \ ?$

I can't seem to eliminate all $x$-variables using $u = x^2 + 1$. I know I can simply compute it by expanding the integrand but I want to do it using substitution in particular.

You can't do this using substitution. Expand the integrand instead.
• Jul 29th 2012, 11:14 AM
Re: Integrating (x² + 1)² using substitution
Quote:

Originally Posted by Prove It
You can't do this using substitution.

How come?
• Jul 29th 2012, 11:27 AM
Prove It
Re: Integrating (x² + 1)² using substitution
Quote:

How come?

I'll rephrase that. You can't perform a u substitution, because doing so requires the substituted function's derivative being a factor of your integrand. If you let u = x^2 + 1, then you need to have a factor of 2x in your integrand as well. You don't, so you can't use u substitution.

You could possibly use a trigonometric or hyperbolic substitution, but doing so is time consuming and pointless, when a perfect square is quick and easy to expand and the resulting integrand easy to integrate using the power rule.