Question

By using the properties of the definite integral, evaluate the integral:

`int_((-pi)/2)^(pi/2) sin^2 x  dx`

Answer 1

Let`I = int_(-pi//2)^(pi//2)  sin^2 x  dx`

`= 2 int_0^(pi//2)  sin^2 x  dx`   ...(i)   ...(∵ sin² x is a function)

Then `I = 2 int_0^(pi//2)  sin^2  (pi/2 - x)  dx`

`= int_0^(pi//2) cos^2 x  dx`  ...(ii)    `[because int_0^a f(x) = int_0^a  f(a - x)  dx]`

On adding equations (i) and (ii)

`2I = 2 int_0^(pi//2) (sin^2  x + cos^2   x)  dx`

`2I = 2 int_0^(pi//2)  1 dx`

`=> 2I = 2 [x]_0^(pi//2)`

`=> 2I = 2 xx pi/2`

Hence, `I = pi/2`