Q 22. - Calculus

Question

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral $\int u dv$.

$\frac{1}{3}x{e}^{3x}-\frac{1}{3}\int e^{3x} dx$

Step-by-Step Solution

Answer

$\begin{array}{rcl}u& =& \frac{x}{3}\\ v& =& e^{3x}\\ du& =& \frac{1}{3}\\ dv& =& 3e^{3x}\\ \int u dv& =& \int x{e}^{3x} dx\end{array}$

Step 1. Given information

Integral is $\frac{1}{3}x{e}^{3x}-\frac{1}{3}\int e^{3x} dx$

Step 2. Explanation

Consider the integral $\frac{1}{3}x{e}^{3x}-\frac{1}{3}\int e^{3x} dx$

$\begin{array}{rcl}u& =& \frac{x}{3}\\ du& =& \frac{1}{3}\\ v& =& e^{3x}\\ dv& =& 3e^{3x}\\ \int u dv& =& \int \left(\frac{x}{3}\right)\left(3e^{3x}\right) dx\\ & =& \int x{e}^{3x} dx\end{array}$