# Integration By Parts

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 doesn’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: 