Question

Integral of cos^5x dx?

Answer 1

`1/80sin(5x)+5/48sin(3x)+5/8sin(x)+C`

Explanation:

`intcos^5(x)dx`

If you know the complex exponential form of `cos(x)` then you could proceed using the following method:

`cos^5(x)=((e^(ix)+e^(-ix))/(2))^5`

Now, expand this with the binomial theorem to get:

`=1/32(e^(i5x)+5e^(i3x)+10e^(ix)+10e^(-ix)+5e^(-3ix)+e^(-5ix))`

Now collect the terms like so:

`=1/32({e^(i5x)+e^(-i5x)}+5{e^(i3x)+e^(-i3x)}+10{e^(-ix)+e^(ix)})`

`=1/16({e^(i5x)+e^(-i5x)}/2+5{e^(i3x)+e^(-i3x)}/2+10{e^(-ix)+e^(ix)}/2)`

`=1/16cos(5x)+5/16cos(3x)+5/8cos(x)`

So from this it may follow that:

`intcos^5(x)dx=int1/16cos(5x)+5/16cos(3x)+5/8cos(x)dx`

Now integrate to get:

`1/80sin(5x)+5/48sin(3x)+5/8sin(x)+C`