What is the integral of cos^2(x)?

What is the integral of cos^2(x)?

• A. (1/2)(x + sin(x)cos(x)) + C
• B. (1/2)(x + sin^2(x)) + C
• C. (1/2)(x + cos^2(x)) + C
• D. (1/2)(x + tan(x)) + C

Answer: (A)

To find the integral of \( \cos^2(x) \), one effective method is to use the trigonometric identity that allows us to express \( \cos^2(x) \) in a more manageable form. The relevant identity is: \[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \] Using this identity, we can rewrite the integral as follows: \[ \int \cos^2(x) \, dx = \int \frac{1 + \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 + \cos(2x)) \, dx \] This integral can be separated: \[ = \frac{1}{2} \left( \int 1 \, dx + \int \cos(2x) \, dx \right) \] Evaluating these integrals, we have: \[ \int 1 \, dx = x \] \[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \] Putting everything together, we find: \[ \

To find the integral of ( \cos^2(x) ), one effective method is to use the trigonometric identity that allows us to express ( \cos^2(x) ) in a more manageable form. The relevant identity is:

[

\cos^2(x) = \frac{1 + \cos(2x)}{2}

]

Using this identity, we can rewrite the integral as follows:

[

\int \cos^2(x) , dx = \int \frac{1 + \cos(2x)}{2} , dx = \frac{1}{2} \int (1 + \cos(2x)) , dx

]

This integral can be separated:

[

= \frac{1}{2} \left( \int 1 , dx + \int \cos(2x) , dx \right)

]

Evaluating these integrals, we have:

[

\int 1 , dx = x

]

[

\int \cos(2x) , dx = \frac{1}{2} \sin(2x)

]

Putting everything together, we find:

[

\