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. $int x^{3}sin(7x)dx$

Explanation:

Step1: Recall integration - by - parts rule

The integration - by - parts formula is $\int u\;dv=uv-\int v\;du$. The general guideline for choosing $u$ is to pick a function that becomes simpler when differentiated.

Step2: Analyze the functions in the integrand

We have two functions in the integrand $x^{3}$ and $\sin(7x)$. The derivative of $x^{3}$ is $3x^{2}$, and the derivative of $\sin(7x)$ is $7\cos(7x)$. Since differentiating $x^{3}$ will reduce the power of the polynomial, we choose $u = x^{3}$.

Answer:

$u=x^{3}$