antidifferentiate sin4 x.

Question

i need help with this question. sin4x is the same as (sinx)^4. we need to antidifferentiate.

Answer 1

cos(2x)=1-2sin²(x), sin²(x)=½(1-cos(2x)), sin⁴(x)=¼(1-2cos(2x)+cos²(2x)).

cos(4x)=2cos²(2x)-1, cos²(2x)=½(cos(4x)+1) so:

sin⁴(x)=¼(1-2cos(2x)+½(cos(4x)+1)=⅛(2-4cos(2x)+cos(4x)+1)=⅛(3-4cos(2x)+cos(4x)).

Each term in this expression can be integrated (antidifferentiated)

The antiderivative of sin⁴(x)=⅛(3x-2sin(2x)+¼sin(4x))+c, where c is a constant.

So ∫sin⁴(x)dx=(1/32)(12x-8sin(2x)+sin(4x))+c.