What is the result of integrating cos^3(x) with respect to x?

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To determine the result of integrating ( \cos^3(x) ) with respect to ( x ), we can use a trigonometric identity and integration by substitution.

The integral can be expressed as:

[

\int \cos^3(x) , dx

]

We can rewrite ( \cos^3(x) ) using the identity ( \cos^2(x) = 1 - \sin^2(x) ):

[

\cos^3(x) = \cos(x) \cdot (1 - \sin^2(x))

]

This gives us:

[

\int \cos^3(x) , dx = \int \cos(x) , dx - \int \cos(x) \sin^2(x) , dx

]

The first part, ( \int \cos(x) , dx ), results in ( \sin(x) ).

For the second part, we use substitution. Let ( u = \sin(x) ), so ( du = \cos(x) , dx ):

[

\int \cos(x) \sin^2(x) , dx = \int u^2 , du = \