Question

integral of (sin(3x))^4dx from 0 to pi

Answer 1

sin^4(3x)=(1/2-1/2cos(6x))^2
1/4-1/2cos(6x)+1/4cos^2(6x)
1/4-1/2cos(6x)+1/4(1/2+1/2cos(12x))
1/4-1/2cos(6x)+1/8+1/8cos(12x)
3/8-1/2cos(6x)+1/8cos(12x)
NOW take the integral of this
3/8x-1/12sin(6x)+1/(96)sin(12x)
since sin(pi)=0 because sin(kpi) k being an integer will be 0
3/8pi will be your answer

Answer 2

NOW check me

Answer 3

(k) being any whole number

Answer 4

\[\int\limits_0^{\pi } \left(\text{Sin}[3 x]^4\right) \, dx=\frac{3 \pi }{8} \]