Why $\int_0^{\pi } {\left(\cos \;t\right)}^{3\ldotp 5}\;\mathrm{dt}=2\int_0^{\frac{\pi }{2}} {\left(\cos \;t\right)}^{3\ldotp 5}\;\mathrm{dt}$?

Question

I can't figure out how this definite integral is transformed like this:

$$\int_0^{\pi } {\left(\cos \;t\right)}^{3\ldotp 5} \;\mathrm{dt}=2\int_0^{\frac{\pi }{2}} {\left(\cos \;t\right)}^{3\ldotp 5} \;\mathrm{dt}$$

I've been studying the integration of even or periodic functions, but can't find anything useful about this particular example. Any clue? Thanks in advance!

Answer 1

Hint

$$\int_0^{2a}f(x)\ dx=\int_0^af(x)\ dx+\int_a^{2a}f(x)\ dx$$

For the last integral, set $2a-x=y$

Can you recognise $f(x), a$ here?