Question

By using the properties of definite integrals,evaluate the integrals: $\int^\frac{\pi}{2}_{-\frac{\pi}{2}} {sin^7\ x} \, dx$

Answer 1

Answer:

0

Step-by-step explanation:

Concept:

A function f(x) is said to be odd function

if f(-x)= - f(x).

If f(x) is odd function, then

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

sin(-x) = - sinx

Now,

$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}{sin^7x}\:dx$

Take,

$f(x)=sin^7x\\\\f(x)=(sinx)^7\\\\f(-x)=(sin(-x))^7\\\\f(-x)=(-sinx)^7\\\\f(-x)=-sin^7x$

f(-x)= - f(x)

Therefore,

f(x) is an odd function

By property of definite integrals,

$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}{sin^7x}\:dx=0$