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« Integration by Substitution
Integration Techniques: (lesson 2 of 4)

Integration by Parts

Theorem:

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

integration by parts

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 = eax

Step 2: Compute du and v

Step 3: Use the formula for the integration by parts

Example 1: Evaluate the following integral

Integration by parts example

Solution:

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

integration by parts step 1

Step 2: Compute du and v

integration by parts step 2

Step 3: Use the formula.

integration by parts step 3

Therefore:

integration by parts solution

Example 2: Evaluate the following integral

Integration by parts example

Solution:

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

integration by parts step 1

Step 2: Compute du and v

integration by parts step 2

Step 3: Use the formula.

integration by parts step 3

The solution is:

integration by parts solution


Exercise 1: Evaluate the following integrals

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

Integration by parts twice

Example 3: Evaluate the following integral

Integration by parts example

Solution:

Let:

Integration by parts step 1

So that

Integration by parts step 2

Therefore:

Integration by parts step 3

We need to perform integration by parts again:

Integration by parts solution

Try yourself

exercisie
answer 1 answer 2 answer 3 answer 4