How to evaluate this integral in terms of a beta function?

Question

I need some help evaluating an integral using beta function.

\begin{equation*} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^4 \phi d\phi \cdot \frac{2\pi}{5} \end{equation*}

I am not sure how to transform lower bound and cosine to fit the equation for beta function. I have to solve it using beta function because that's what the exercise says.

Answer 1

Hint.

For any even (integrable) function $f$, $$ \int_{-a}^af(x)dx=2\int_0^{a}f(x)dx $$

Also, one of the various ways to write the Beta function is:

$$ {\displaystyle {\begin{aligned}\mathrm {B} (x,y)&=2\int _{0}^{\pi /2}(\sin \theta )^{2x-1}(\cos \theta )^{2y-1}\,d\theta \\[6pt]& \end{aligned}}} $$