How do you integrate stuff like cos2xdx or sin10xdx??

Question

Do I follow a formula or is there more than one step? I'M CONFUSED!!!

Answer 1

The way we do it is look it up in tables.

However, before we got to that point, we had to understand how it is really done, just so that we know when to trust the table or not.

In these cases, one uses substitution.

In the integration of cos(2x) dx,
one replaces 2x by another variable (say u = 2x)
which then leads to du = 2 dx (or dx = du / 2)

Rewrite the function:

cos(u)du/2 = (1/2) cos u du
The (1/2) can be moved outside the integral (it is a simple factor) and you are left with a simple integral cos u du
which is sin(u) + C
Don't forget to replace the u by 2x AND multiply the result by the (1/2) you moved outside.

(1/2) sin(2x) + C

Answer 2

You have to use u-substitution for these types of integration problems.
For instance, on the first problem take u = (2x), and du = 2dx. This means that, solving for dx, dx= 1/2 du. Now, remember to put the 1/2 outside of the integral as a constant:

int (cos2x) dx
= 1/2 int (cos u) du
= 1/2 (sin u) + C
= 1/2 (sin 2x) + C

Remember to plug back in your u in the final step, and add the constant C after integrating!
Follow almost the same procedure for sin10xdx. Do you understand now? I hope I helped!