We saw that substitution rule helps us solve chain rule related derivatives with relative ease. Integration by parts is a method used to solve product rule related derivatives.

**Integration
by Parts: **For functions in the form ,
the integral

Often, we choose to represent f(x) = u and g(x) = v to make the formula more compact and easy to remember:

**Example:
**Find

For this example, we can select u = x and dv = sin(x), then du = 1 and v = -cos(x). Now we plug this into our formula to get

When we are selecting u and dv, we need to be careful with our selections. You might notice an alternative option would be to choose u = sin(x) and dv = x. This would give us du = cos(x) and v = . This would make our formula:

This formula results in an integral that is just as hard as the original, meaning it doesnâ€™t benefit us at all. When we are picking the u value, there is a general rule that helps, known as LIATE. LIATE stands for Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential. It is the order of preference for picking a u value. For example, x is algebraic and sin(x) is trigonometric, since algebraic comes first, we prefer it over sin(x).

**Example:
**Find

Following the idea of LIATE, we have an algebraic term, and an exponential term. We prefer the algebraic term, meaning , and . This gives us and , meaning we get the integral:

Notice that is still relatively hard to solve. In cases like this, we need to apply integration by parts again. We can pick u = 2t, . This gives us du = 2 and , meaning:

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