Question

in using the technique of integration by parts, you must carefully choose which expression is $u$. for the following problem, use the guidelines in this section to choose $u$. do not evaluate the integral. $int x^{3}ln(6x)dx$

Explanation:

Step1: Recall integration - by - parts rule

The integration - by - parts formula is $\int u\;dv=uv-\int v\;du$. We want to choose $u$ such that its derivative $du$ is simpler than $u$, and $dv$ such that its antiderivative $v$ can be found relatively easily.

Step2: Analyze the functions

We have two functions in the integrand: $x^{3}$ and $\ln(6x)$. The derivative of $\ln(6x)$ is $\frac{1}{x}$, which is a simpler function. The antiderivative of $x^{3}$ is $\frac{1}{4}x^{4}$. So, if we choose $u = \ln(6x)$ and $dv=x^{3}dx$, we can apply the integration - by - parts formula.

Answer:

$u=\ln(6x)$