Question

How do you find the definite integral for: `e^sin(x) * cos(x) dx` for the intervals `[0, pi/4]`?

Answer 1

Use a `u`-substitution to get `int_0^(pi/4) e^sinx*cosxdx=e^(sqrt(2)/2)-1`.

Explanation:

We'll begin by solving the indefinite integral and then deal with the bounds.

In `inte^sinx*cosxdx`, we have `sinx` and its derivative, `cosx`. Therefore we can use a `u`-substitution.

Let `u=sinx->(du)/dx=cosx->du=cosxdx`. Making the substitution, we have:
`inte^udu`
`=e^u`

Finally, back substitute `u=sinx` to get the final result:
`e^sinx`

Now we can evaluate this from `0` to `pi/4`:
`[e^sinx]_0^(pi/4)`
`=(e^sin(pi/4)-e^0)`
`=e^(sqrt(2)/2)-1`
`~~1.028`