Question

Evaluate the following using properties of definite integral:

`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x)  "d"x`

Answer 1

Using the property

`int_0^"a" "f"(x)  "d"x = int_0^"a" "f"("a" - x)  "d"x`

Let I = `int_0^(pi/2)  (sin^7x)/(sin^7x + cos^7x)  "d"x` ........(1)

I = `int_0^(pi/2) (sin^7(pi/2 - x))/(sin^7(pi/2 - x) + cos^7(pi/2 - x))  "d"x`

I = `int_0^(pi/2) (cos^7x)/(cos^7x + sin^x)  "d"x`  .........(2)

Adding (1) and (2)

I + I = `int_0^(pi/2) (sin^7x + cos^7x)/(sin^7x + cos^7x  "d"x`

2I  `int_0^(pi/2)  "d"x`

2I = `[x]_0^(pi/2) = [pi/2 - 0]`

2I = `pi/2`

⇒ I = `pi/4`