how do you integrate cos(2x)?

Question

Can someone show me step by step the integral/antiderivative of cos(2x)? I know the answer is (1/2)sin(2x) but I don't know how to get that. Thanks

Answer 1

Well, its easy. first you can use u-substitution and substitute 2x for u. then differentiate 2x and u. thus, du=2dx. use that in the original equation.
Thus (int)cos(2x)dx= 1/2(int)cos(u)du
this will be equal to 1/2(-sin(u)) now you can put back u=2x in the answer.
thus the final answer is -0.5sin2x.
in case you were wondering where the 0.5 came from, well, when you differentiated u=2x, the answer came as du=2dx.
as there was no 2dx in the original eq. only dx, thus we had to put an extra 2 in the equation, and to undo that step, put a 1/2 on the outside.
trust me, its looks complicated but its easy

Answer 2

The usual way. Let u=2x and du=2dx.
Then we have ader((1/2)cos udu).
We know that cos u integrates to sin u, so the answer is (1/2) sin (2x)

Answer 3

you should integrate that cos fucntion first.. which will give you sin2x..

then next you integrate the 2x.. which will give you 1/2.

so combine them and you'll get (1/2)sin(2x)

Answer 4

it follows simply as elementary result from (sin(ax))' = acos(ax)
so integral cos(ax) = (1/a) sin(ax)
here a =2