Question

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

Answer 1

Answer:

Step-by-step explanation:

Concept:

1.A function f(x) is said to be even function

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

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

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

3.Reduction formula:

$\int\limits^{\frac{\pi}{2}}_{0}{sin^2x}\:dx=(\frac{n-1}{n})(\frac{n-3}{n-2}).....(\frac{1}{2}).\frac{\pi}{2}$ if n is even

Now,

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

Take,

$f(x)=sin^2x\\f(x)=(sinx)^2\\f(-x)=(sin(-x))^2\\f(-x)=(-sinx)^2\\f(-x)=sin^2x\\f(-x)= f(x)$

Therefore,

f(x) is an even function

By property of definite integrals,

$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}{sin^2x}\:dx\\\\=2(\int\limits^{\frac{\pi}{2}}_{0}{sin^2x}\:dx)\\\\=2(\frac{1}{2}.\frac{\pi}{2})\\\\=\frac{\pi}{2}$

Answer 2

Answer:

Concept:

1.A function f(x) is said to be even function

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

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

3.Reduction formula:

if n is even

Now,

Take,

Therefore,

f(x) is an even function

By property of definite integrals,

Step-by-step explanation: