Class 17: §6.1 Integration By Parts

Today’s Plan

Roll Call
Start learning to evaluate more integrals!

Today’s Assignments

We’ll complete material for Learning Problems 07 today and also explore it further tomorrow in Groupwork 05.

Warm-up Activity

Setup an integral that represents the volume generated by rotating the region bounded by the curves $y=\sin x\qquad \text{and} \qquad y=0$ from $x=0$ to $x=\frac{\pi}{2}$ .

Let’s investigate the Product Rule!

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.

Integration by Parts

For two differentiable functions $u$ and $v$ (each depending on $x$ ) $\int u\;dv = uv - \int v \;du.$

For definite integrals evaluated from limits $a$ to $b$ , boundary evaluations are calculated as $\int_a^b u\;dv = uv\Big|_a^b - \int_a^b v \;du.$

Example 1

Use Integration by PartsFor two differentiable functions $u$ and $v$ (each depending on $x$ ) $\int u\;dv = uv - \int v \;du.$ For definite integrals evaluated from limits $a$ to $b$ , boundary evaluations are calculated as $\int_a^b u\;dv = uv\Big|_a^b - \int_a^b v \;du.$ to evaluate the integral. $\int x \sin(x)~dx.$

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.

Useful Mnemonic

The following chart can be useful when frequently integrating and differentiating $\sin(x)$ and $\cos(x)$ , as is common with integration by parts problems. Writing this out can speed up keeping signs straight when switching constantly back and forth.

$\begin{matrix} \color{gray}{\uparrow} &\quad&\sin (x) &\quad& \color{gray}{\frac{d}{dx}(\_)} \\ \color{gray}{\uparrow} && \cos (x) && \color{gray}{\downarrow} \\ \color{gray}{\uparrow} && - \sin(x) && \color{gray}{\downarrow} \\ \color{gray}{\int(\_)}\;dx && -\cos(x) && \color{gray}{\downarrow} \end{matrix}$

Example 2

Find $\int \ln(x)~dx.$

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.

Prompt

You only have two choices for $u$ and $dv$ in the integration by parts formula. One of them makes no forward progress (why?). Therefore there is only one reasonable first step to this problem. If you find that, you’re mostly done already.

Example 3 (The tricky one)

Find $\int e^x \sin(x)~dx.$

Completed on Board

We will work through the solution as a class, with the instructor leading and students contributing key steps, ideas, and justifications throughout.

Examples 4 and 5

Find $\int t^2 e^t~dt$ and $\int_0^1 \tan^{-1}(x)~dx.$

Work then Guided Solution

Students will begin with individual or collaborative work time, followed by a guided solution on the board developed through student input. Be prepared to explain your reasoning and contribute to each step of the process.

Extra Practice

Go back and complete the warm-up activity to find the volume of the solid of revolution.