Question

How do you integrate `int x cos 3 x dx` using integration by parts?

Answer 1

Step by step working is shown below on how to go about integration by parts.

Explanation:

`intxcos(3x)dx`

Integration by parts rule is

`int udv = uv-int vdu`

First we select `u`
Using a mnemonic ILATE which stands for Inverse, Logarithmic, Algebraic, Trigonometric and Exponential we select `u` in the preferred order given by ILATE.

In our problem, we can select `u=x` this is because Algebraic function is preferred above the Trigonometric.

`u=x`
Differentiating with respect to `x`
`du = dx`

Once `u` is selected remaining terms for the `dv`

`dv=cos(3x)dx`

To find `v` we have to integrate this function.

`v=int dv = int cos(3x)dx`

`v=sin(3x)/3`

Using the integration part rule

`int udv = uv-int vdu`

`intxcos(3x)dx = 1/3xsin(3x) - int(sin(3x)/3)dx`

`intxcos(3x)dx = 1/3xsin(3x) +1/9cos(3x)+C`