Question

How do you find `-\int \cos ( e ^ { u } ) e ^ { 4} d u`?

Answer 1

`-int\ cos(e^u)e^4\ du=-e^4Ci(e^u)+C`

Explanation:

First, we bring out the constant, `e^4`:
`-e^4int\ cos(e^u)\ du`

Next, we will introduce a substitution with `t=e^u`. The derivative of `t` is `e^u=t`, so we divide by `t` to integrate with respect to `t`:
`-e^4int\ cos(e^u)\ du=-e^4int\ cos(t)/t\ dt`

This integral that we have now is called the Cosine integral. It cannot be represented using elementary functions, but we can label it using `Ci(x)`.

In our case, we get:
`-e^4int\ cos(t)/t\ dt=-e^4Ci(t)+C=-e^4Ci(e^u)+C`