How do you find the antiderivative of sin^4(x)?

1 Answer
Apr 29, 2015

You could use power reduction formulas (double angle)

Because cos 2 theta = 1-2sin^2 theta = 2cos theta -1, we get

sin^2theta = 1/2 (1-cos2 theta)
and
cos^2theta = 1/2 (1+cos2 theta)

int sin^4x dx = int (sin^2x)^2 dx

color(white)"sssssssssss" = int [1/2(1-cos2x)]^2 dx

color(white)"sssssssssss" = 1/4 int (1-2cos2x+ cos^2 2x) dx

color(white)"sssssssssss" = 1/4 int (1-2cos2x+ 1/2(1+cos4x)) dx

color(white)"sssssssssss" = 1/4 int (3/2-2cos2x+ 1/2 cos4x) dx

color(white)"sssssssssss" = 1/4 (3/2 x - sin2x+ 1/8 sin4x) +C

color(white)"sssssssssss" = 3/8 x- 1/4 sin2x+ 1/32 sin4x +C