Question

`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.

Options

• `pi/4`

• `pi/2`

• π

• 2π

Answer 1

`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = `pi/4`.

Explanation:

Let I = `int_0^(pi/2) 1/(1 + cos^3x) "d"x` 

∴ I = `int_0^(pi/2) (sin^3x)/(sin^2x + cos^2x) ""x`  ......(i)

∴ I = `(cos^3x)/(cos^3x + sin^3x) "d"x`  ......(ii) `[because int_0^"a" "f"(x)"d"x = int_0^"a" "f"("a" - x)"d"x]`

Adding (i) and (ii), we get

2I = `int_0^(pi/2) (sin^3x + cos^x)/(sin^3x  cos^3x) "d"x`

∴ 2I = `int_0^(pi/2) 1* "d"x`

∴ I = `1/2[x]_0^(pi/2)`

= `1/2(pi/2 - 0)`

∴ I = `pi/4`