How to choose u for integration by parts?

Choose u and v:

Differentiate u: sin(x)' = cos(x)

Integrate v: ∫ex dx = ex

Now put it together:

It looks worse, but let us persist! To find ∫cos(x) ex dx we can use integration by parts again:

Choose u and v:

Differentiate u: cos(x)' = -sin(x)

Integrate v: ∫ex dx = ex

Now put it together:

Simplify:

Now we have the same integral on both sides (except one is subtracted) ...

... so we can bring the right hand integral over to the left and we get: