Question

How do you integrate `int cos^3x dx`?

Answer 1

I found: `sin(x)-(sin^3(x))/3+c`

Explanation:

Have a look:

Answer 2

`sin(x)cos(x)^2+2/3sin(x)^3+C`

Explanation:

`d/(dx)(sin(x)cos(x)^2)=cos(x)^3-2sin(x)^2cos(x)`

then

`int cos(x)^3dx = sin(x)cos(x)^2+2intsin(x)^2cos(x)dx`

but

`intsin(x)^2cos(x)dx = int(1/3 d/(dx)sin(x)^3)dx`

finally

`int cos(x)^3dx = sin(x)cos(x)^2+2/3sin(x)^3+C`

Answer 3

`int cos^3 x d x=sin x-1/3sin^3 x+C`

Explanation:

`"a different way..."`

`"use reduction formula"`

`int cos^n x d x=(n-1)/n int cos^(n-2) x d x+(cos^(n-1)x*sin x)/n`

`"use n=3"`

`int cos^3 x d x=(3-1)/3 int cos^(3-2)x +(cos^(3-1)x*sin x)/(3)`

`int cos^3 x d x=2/3int cos x d x+(cos^2 x*sin x)/3`

`cos^2x=1-sin^2x`

`int cos^3 x d x=2/3 sin x+((1-sin^2 x)*sin x)/3`

`int cos^3 x d x=2/3 sin x+(sin x-sin^3 x)/3`

`int cos^3 x d x=(2 sin x+sin x-sin^3 x)/3`

`int cos^3 x d x=(3sin x-sin^3 x)/3`

`int cos^3 x d x=sin x-1/3sin^3 x+C`