How do you evaluate the integral of int (sin^4x)dx?

1 Answer
Jan 17, 2016

Definitively the fastest way to do that, is to use euler formula

sin^4(x) = (1/(2i)(e^(ix)-e^(-ix)))^4

(1/(2i))^4 = (1/2^4) because i^4 = 1

expand with binomial theorem

e^(4ix)-4e^(3ix)e^(-ix)+6e^(2ix)*e^(-2ix)-4e^(ix)*e^(-3ix)+e^(-4ix)

=1/2^4((e^(4ix)+e^(-4ix))-4(e^(2ix)+e^(-2ix))+6)

=1/2^4(2cos(4x) - 8cos(2x) + 6)

so

int sin^4(x) dx = 1/2^4int2cos(4x)-8cos(2x)+6dx

which is

1/2^4[1/2sin(4x)-4sin(2x)+6x]