Question

Evaluate the following integrals using properties of integration:

`int_0^pi sin^4 x cos^3 x  "d"x`

Answer 1

`int_0^pi sin^4 x cos^3 x  "d"x`

f(x) = sin⁴x cos³x

f(2π – x) = sin⁴(2π – x) cos³(2π – x)

= sin⁴x cos³x

f(2π – x) = f(x)

if `"f"(2"a" - x) = "f("x)` then `int_0^(2"a") "f"(x)  "d"x`

= `2 int_0^"a" "f"(x)  "d"x`

I = `2 int_0^pi sin^4 x cos^3 x  "d"x`

t = sin x

dt = cox dx

x	0	`pi`
t	0	0

 ` 2int_0^pi  sin^4x(1 - sin^2x) cosx  "d"x`

Limit from 0 to π tends to 0 to 0

∴ Integral value = 0

∴ `int_0^pi sin^4 x cos^3x  "d"x` = 0