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. $int x^{6}e^{7x}dx$

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 we can integrate it to get $v$.

Step2: Analyze the functions in the integrand

We have two functions in the integrand: $x^{6}$ and $e^{7x}$. The derivative of a polynomial function $x^{n}$ ($n$ is a non - negative integer) becomes a polynomial of lower degree. The derivative of $e^{ax}$ ($a$ is a constant) is $ae^{ax}$, which is of the same form. So, if we choose $u = x^{6}$, its derivative $du=6x^{5}dx$ is a simpler polynomial function.

Answer:

$u = x^{6}$