What is the integral of (cos⁡x)2?

Question

sirpsta3u · Accepted Answer

Explanation:
We will use the cosine double-angle identity in order to rewrite

\cos^{2} x

. (Note that

\cos^{2} x = {(\cos x)}^{2}

, they are different ways of writing the same thing.)

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

This can be solved for

\cos^{2} x

:

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

Thus,

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

Split up the integral:

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

The second integral is the "perfect integral:"

\int d x = x + C

= \frac{1}{2} \in \cos (2 x) d x + \frac{12}{x}

The constant of integration will be added upon evaluating the remaining integral.
For the cosine integral, use substitution. Let

u = 2 x

, implying that

d u = 2 d x

Multiply the integrand 2 and the exterioe of the integral by

\frac{12}{}

.

= \frac{14}{\int} 2 \cos (2 x) d x + \frac{12}{x}

Substiute in u and du:

= \frac{14}{\int} \cos (u) d u + \frac{12}{x}

Note that

\int \cos (u) d u - \sin (u) + C

= \frac{14}{\sin (u)} + \frac{12}{x} + C

Since

u = 2 x

:

= \frac{14}{\sin (2 x)} + \frac{12}{x} + C

Note that this can be many different ways, since

\sin (2 x) = 2 \sin x \cos x

.

einfachmoipf · Accepted Answer

Use the half angle formula to write cos2(x) as 1+cos⁡(2x)2  ∫1+cos⁡(2x)2dx  Since 1/2 is constant with respect to x, take out 1/2 of the integral.  12∫1+cos⁡(2x)dx  We expand the integral into several integrals.  12∫1dx+∫cos⁡(2x)dx  Since 1 is constant with respect to x, move 1 out of the integral.  12(x+C+∫cos⁡(2x)dx)  Let be u=2x. Then du=2dx, hence 12du=dx. Rewrite using u and du.  Combine cos⁡(u) and 12.  12x+C+∫cos⁡(u)2du  Insofar as 12 is constant with repect to u, take out 12 from the integral.  12(x+C+12∫cos⁡(u)du)  12x+12sin⁡(2x)+C  So,  12x+14sin⁡(2x)+C

RizerMix · Accepted Answer

It&#039;s so simple, cool

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

Answered question

Answer & Explanation

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