Theorem:

The formula for the method of integration by parts is given by:

Four steps to use this formula:

Step 1: Identify u and dv. Priorities for Choosing u are: 1. u = lnx 2. u = xⁿ 3. u = e^ax

Step 2: Compute du and v

Step 3: Use the formula for the integration by parts

Example 1: Evaluate the following integral

Solution:

Step 1: In this example we choose u = x and dv will be everything else that remains.

Step 2: Compute du and v

Step 3: Use the formula.

Therefore:

Example 2: Evaluate the following integral

Solution:

Step 1: In this example we choose u = ln x (first priority) and dv = x dx.

Step 2: Compute du and v

Step 3: Use the formula.

The solution is:

Exercise 1: Evaluate the following integrals

Level 1

Level 2

Integration by parts – twice

Example 3: Evaluate the following integral

Solution:

Let:

So that

Therefore:

We need to perform integration by parts again:

Try yourself