how do I solve the integral sin^4(5x)dx from 0 to pi/5?

Answer 1

Integrate sin^4(5x) with respect to x [from 0 to π/5].

First rearrange the expression to make it easier to integrate.

sin^4(5x) = sin²(5x)[1 - cos²(5x)] = sin²(5x) - sin²(5x)[cos²(5x)]
= (1/2)[1 - cos(10x)] - (1/2)sin(10x)

Now we can integrate.

∫sin^4(5x)dx
= ∫{(1/2)[1 - cos(10x)] - (1/2)sin(10x)}dx
= x/2 - (1/20)sin(10x) + (1/20)cos(10x) [eval from 0 to π/5]
= (1/20){10x - sin(10x) + cos(10x)}
= (1/20){10π/5 - sin(10π) + cos(10π)} - (1/20){0 - sin(0) + cos(0)}
= (1/20){2π - 0 + 1} - (1/20){0 - 0 + 1}
= (1/20)(2π) = π/10