Integration Techniques: (lesson 2 of 4)

## Integration by Parts

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 = x*^{n} 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

### Integration by parts twice

Example 3: Evaluate the following integral

Solution:

Let:

So that

Therefore:

We need to perform integration by parts again: