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How to simplify sin^4x/4 + cos^4x/4?
The integration of sin^3xcosx + sinxcos^3x dx
the answer is 1/2sin^2x + C on my book
But i dont know how the answer became life that im stuck in sin^4x\4 + cos^4x/4
1 Answer
- ?Lv 68 years agoFavorite Answer
sin^3xcosx + sinxcos^3x = sinxcosx(sin^2(x) + cos^2(x))
we know that sin^2(x) + cos^2(x)=1, then
sin^3xcosx + sinxcos^3x = sinxcosx = (1/2)sin(2x)
∫(1/2)sin(2x)= -cos(2x)/4 + C {not the same answer}
Simplification
sin^4(x)/4 + cos^4(x)/4 = (sin^2(x) + cos^2(x))^2 - 2sinxcosx
sin^2(x) + cos^2(x) = 1 and 2sinxcosx=sin(2x), therefore
sin^4(x)/4 + cos^4(x)/4 = 1 -sin(2x)
I hope this helps
