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

Jun 27, 2016

Use a $u$-substitution to get ${\int}_{0}^{\frac{\pi}{4}} {e}^{\sin} x \cdot \cos x \mathrm{dx} = {e}^{\frac{\sqrt{2}}{2}} - 1$.

#### Explanation:

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

In $\int {e}^{\sin} x \cdot \cos x \mathrm{dx}$, we have $\sin x$ and its derivative, $\cos x$. Therefore we can use a $u$-substitution.

Let $u = \sin x \to \frac{\mathrm{du}}{\mathrm{dx}} = \cos x \to \mathrm{du} = \cos x \mathrm{dx}$. Making the substitution, we have:
$\int {e}^{u} \mathrm{du}$
$= {e}^{u}$

Finally, back substitute $u = \sin x$ to get the final result:
${e}^{\sin} x$

Now we can evaluate this from $0$ to $\frac{\pi}{4}$:
${\left[{e}^{\sin} x\right]}_{0}^{\frac{\pi}{4}}$
$= \left({e}^{\sin} \left(\frac{\pi}{4}\right) - {e}^{0}\right)$
$= {e}^{\frac{\sqrt{2}}{2}} - 1$
$\approx 1.028$