Question

By using the properties of definite integrals,evaluate the integrals: $\int^\frac{\pi}{2}_{-\frac{\pi}{2}} {cos^5\ x} \, dx$

Answer 1

Answer:

Step-by-step explanation:

Formula used:

1.If f(x) is an even function then,

$\int\limits^{a}_{-a}{f(x)}\:dx=2\int\limits^{a}_{0}{f(x)}\:dx$

2.Reduction formula:

$\int\limits^{\frac{\pi}{2}}_{\frac{-\pi}{2}}{cos^nx}\:dx=\frac{n-1}{n}.\frac{n-3}{n-2}.........\frac{2}{3}.1$

if n is odd

$f(x)=cos^5x\\\\f(-x)=[cos(-x)]^5\\\\f(-x)=[cosx]^5\\\\f(-x)=cos^5x\\\\f(-x)=f(x)$

Therefore, f(x) is an even function.

By property of definite integrals,

$\int\limits^{\frac{\pi}{2}}_{\frac{-\pi}{2}}{cos^5x}\:dx\\\\=2\int\limits^{\frac{\pi}{2}}_0{cos^5x}\:dx\\\\=2.(\frac{4}{5}).(\frac{2}{3}).1\\\\=\frac{16}{15}$