Question

How do you integrate `e^(sinx) cosx dx`?

Answer 1

`e^(sin(x))+C`

Explanation:

You can solve the integral using a u-substitution

Let `u=sin(x)`

Differentiating we get

`du=cos(x)dx`

Make the subtitution

`int e^udu`

integrating we get `e^u`

Now back substitute for `u`

`e^(sin(x))+C`

Answer 2

if you recognise the result
`" "d/(dx)(e^f(x))=f'(x)e^((f(x))" "`

you can integrate this directly.

Explanation:

`" "d/(dx)(e^sinx)=cosxe^(sinx)" "`

so `intcosxe^(sinx)dx" "=e^sinx+C`