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

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

• A. (3/4)sin(x) + (1/4)sin^3(x) + C
• B. (1/3)sin^3(x) + C
• C. (1/2)sin(x) + (1/2)sin^3(x) + C
• D. (1/4)sin(x) + (3/4)sin^3(x) + C

Answer: (A)

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 = \

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 = \