Question

What's the antiderivative of (sin3x)^2 do with upper limit of Pi/2 and lower limit of 0?

Answer 1

`I=int_0^(pi/2)sin^2(3x)dx=pi/4`

Explanation:

I assume you mean the integral

`I=int_0^(pi/2)sin^2(3x)dx`

Remember the trig identity `color(blue)(sin^2(a)=1/2-1/2cos(2a))`

`I=int_0^(pi/2)1/2-1/2cos(6x)dx`

`color(white)(I)=[1/2x-1/12sin(6x)]_0^(pi/2)`

`color(white)(I)=(1/2*pi/2-1/12sin(3pi))-(0-1/12sin(0))`

`color(white)(I)=pi/4`