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:
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).
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: