Find the integral of x⁴(1+x⁵)⁵

by RoRi

October 31, 2015

Evaluate the following integral:

$\int x^4 (1+x^5)^5 \, dx.$

We make the substitution

$u = 1+x^5 \qquad \implies \qquad du = 5x^4 \, dx.$

Then,

$\begin{align*} \int x^4 (1+x^5)^5 \, dx &= \frac{1}{5} \int 5x^4(1+x^5)^5 \, dx \\ &= \frac{1}{5} \int u^5 \, du \\ &= \frac{1}{5} \left( \frac{u^6}{6} \right) \\ &= \frac{1}{30} (1+x^5)^6. \end{align*}$

Stumbling Robot

Find the integral of x⁴(1+x⁵)⁵

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