It's me or integrating sin^6(x) takes forever

i am at a point where i have to do 10 trigonometric substitutions.

wtf, are they expecting me to do?

are they ******* insane?

"are they ******* insane?" Probably. Stupid is more like it.

It is a sign of a poor teacher to give long, meaningless calculations.

The point can be brought home by integrating sin^3(x), much shorter and carrying the same message.

You can do it a little faster by playing with complex numbers.

sin^6(x)
= [ exp(ix) - exp(-ix) ]^6 / (2i)^6
= [exp(6ix) - 6exp(4ix) + 15exp(2ix) - 20 + 15exp(-2ix) - 6exp(-4ix) + exp(-6ix)] / (-64)
= [ 2 cos(6x) - 12 cos(4x) + 30 cos(2x) - 20 ] / (-64)
= [ cos(6x) - 6 cos(4x) + 15 cos(2x) - 10 ] / (-32)

The integral is

[ sin(6x)/6 - (3/2) sin(4x) + (15/2) sin(2x) - 10x ] / (-32)

you could do an integration by parts [ sin x ]^6 dx = [ sinx ]^5 { - cos x } / 6 +[5/6] { [sin x]^4 dx }

integrating the last by parts will take you to integration of sin² x { = [1-cos 2x]/2 dx }


u = sin^5 x , dv = sin x dx to start...try it

no it's not you and yeah it's kind of crazy but good practice i guess. There are tables you could use but they want you to grind it out when you're first learning it. I know it sucks...

it takes a while. don't worry, you're probably doing it right. it'll take like a full page or more.