Find the integral of xn log (ax)

Evaluate the following integral:

$\int x^n \log (ax) \, dx.$

Here we want to use integration by parts. To that end we define,

$\begin{align*} u &= \log (ax) & du &= \frac{1}{x} \, dx \\ dv &= x^n \, dx & v &= \frac{x^{n+1}}{n+1}, \end{align*}$

if $n \neq -1$ .

Therefore, integrating by parts we obtain the integral for all $n \neq -1$ .

$\begin{align*} \int x^n \log (ax) \, dx &= \int u \, dv \\ &= uv - \int v \, du \\ &= \frac{x^{n+1}}{n+1} \log |ax| - \frac{1}{n+1} \int x^n \, dx \\ &= \frac{x^{n+1}}{n+1} \log |ax| - \frac{x^{n+1}}{(n+1)^2} + C \end{align*}$

Now, for the case that $n = -1$ we have

$\int x^n \log (ax) \, dx = \int \frac{\log (ax)}{x} \, dx.$

Since $\frac{1}{x} \, dx$ is the derivative of $\log (ax)$ we make the substitution $u = \log (ax)$ and $du = \frac{dx}{x}$ to obtain

$\begin{align*} \int \frac{\log(ax)}{x} \, dx &= \int u \, du \\ &= \frac{1}{2} u^2 + C \\ &= \frac{1}{2} (\log |ax|)^2 + C. \end{align*}$

Stumbling Robot

Find the integral of xⁿ log (ax)

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