Integration by Parts

We learned the fact that the Integration is the inverse of Differentiation. For Every differentiation rule there is a corresponding integration rule. Like the Substitution Rule for integration corresponds to the Chain Rule for differentiation. Now, the rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts.

The Formula

The method of integration by parts is based on the product rule for differentiation:

The Steps

• The ability to choose u and dv correctly.
• If the choice is right, the new integral that  you obtain  is simpler than the original one.
• Integrate using the Integration by parts formula
• Check the answer by differentiating.

The Examples

1.

Let:               u = x                           dv = e^x dx
then:           du = dx                           v = e

Solution:

Check by differentiating:

2.

Let:               u = x                           dv = sin (x) dx
then:           du = dx                           v = -cos (x)

Solution:

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