Question

How do you evaluate the definite integral `int e^(sin(x))*cosx dx` from `[0,pi]`?

Answer 1

`0`

Explanation:

We have

`int_0^pie^sinxcosxdx`

If we let `u=sinx`, then `du=cosxdx`.

We will want to change the bounds by plugging the current bounds into our `u=sinx` expression.

`u(0)=sin(0)=0`
`u(pi)=sin(pi)=0`

Notice that something fairly odd has happened--both our bounds have become `0`. We have the definite integral

`=int_0^0e^udu`

When both the bounds of an integral are the same, the integral's value is `0`.

Recall that the integral just finds the area under a curve--there is no area if we are traveling from `x=0` to `x=0`, since we haven't moved at all. This is essentially a rectangle with width `0`.