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:
1 thought on “Integration By Parts”