Expert answers to "Evaluate the integral. ∫cos24xdx"

Dottie Parra

Answered question

2021-11-05

Evaluate the integral.

\int \cos^{2} 4 x d x

avortarF

Skilled2021-11-06Added 113 answers

Step 1
We use the double angle formula

\cos (8 x) = 2 \cos^{2} (4 x) - 1

2 \cos^{2} (4 x) = \cos (8 x) + 1

\cos^{2} (4 x) = \frac{\cos (8 x) + 1}{2}

Step 2
Then we use that in the given integral

\int \cos^{2} (4 x) d x

= \int \frac{\cos (8 x) + 1}{2} d x

= \frac{1}{2} \int [\cos (8 x) + 1] d x

= \frac{1}{2} [\frac{\sin (8 x)}{8} + x] + C [∵ \int \cos (m x) d x = \frac{\sin (m x)}{m}]

= \frac{\sin (8 x)}{16} + \frac{x}{2} + C

Answer:

\frac{\sin (8 x)}{16} + \frac{x}{2} + C

C=integrating constant

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