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

1 Answer
Jan 10, 2016

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