Integration by Parts

In some cases an integral can not be solved using u-substitution or the standard integration rules. In this case another method called integration by parts can be used.

Integration by parts

Integrals that can be easily solved using Integration by Parts consist of 2 functions that are multiplied. Where one of the two functions becomes a simpler function after it is differentiated. The general form of these integrals is shown below:

In order to apply Integration by Parts the form as shown above must be recognized. One function is chosen to be an integral of $f(x)$ and the other function is just equal to $g(x)$. The function that becomes simpler after differentiating the function is chosen to be equal to $F(x)$ and the other part is chosen to be equal to $g(x)$. When this is done, it is needed to apply the general rule for integration by parts as shown below.

Example

The integral below must be solved using Integration by Parts.

The necessary form for Integration by Parts is recognized. In this case $F(x) = x$, since this part becomes simpler after it is differentiated. And $g(x) = sin(x)$.

1. $F(x) = x$ so $f(x) = 1$ (Differentiation)
2. $g(x) = sin(x)$ so $G(x) = -cos(x)$ (Integration)
3. $\int{}{} xsin(x)dx = -xcos(x) - \int{}{} -cos(x)dx$ (Applying Integration by Parts)
4. $\int{}{} xsin(x)dx = -xcos(x) - (-sin(x))$ (Working out the second integral)
5. $\int{}{} xsin(x)dx = -xcos(x) + sin(x) + c$ (Solution)

Now is shown how most integrals can be solved using one of the 3 methods explained in this series consisting of the standard integration rules, u-substitution and Integration by Parts.