What is the integral of ${(cos x)}^{2}$ $(cosx)^2$ ?

Question

What is the integral of ${(cos x)}^{2}$ $(cosx)^2$ ?

1 Answer

Impact of this question

200129 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-12T16:27:41

$\frac{1}{4} sin (2 x) + \frac{1}{2} x + C$ $1/4sin(2x)+1/2x+C$

Explanation:

We will use the cosine double-angle identity in order to rewrite ${cos}^{2} x$ $cos^2x$ . (Note that ${cos}^{2} x = {(cos x)}^{2}$ $cos^2x=(cosx)^2$ , they are different ways of writing the same thing.)

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

This can be solved for ${cos}^{2} x$ $cos^2x$ :

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

Thus,

$\int {cos}^{2} x d x = \int \frac{cos (2 x) + 1}{2} d x$ $intcos^2xdx=int(cos(2x)+1)/2dx$

Split up the integral:

$= \frac{1}{2} \int cos (2 x) d x + \frac{1}{2} \int d x$ $=1/2intcos(2x)dx+1/2intdx$

The second integral is the "perfect integral:" $\int d x = x + C$ $intdx=x+C$ .

$= \frac{1}{2} \int cos (2 x) d x + \frac{1}{2} x$ $=1/2intcos(2x)dx+1/2x$

The constant of integration will be added upon evaluating the remaining integral.

For the cosine integral, use substitution. Let $u = 2 x$ $u=2x$ , implying that $d u = 2 d x$ $du=2dx$ .

Multiply the integrand $2$ $2$ and the exterior of the integral by $\frac{1}{2}$ $1/2$ .

$= \frac{1}{4} \int 2 cos (2 x) d x + \frac{1}{2} x$ $=1/4int2cos(2x)dx+1/2x$

Substitute in $u$ $u$ and $d u$ $du$ :

$= \frac{1}{4} \int cos (u) d u + \frac{1}{2} x$ $=1/4intcos(u)du+1/2x$

Note that $\int cos (u) d u = sin (u) + C$ $intcos(u)du=sin(u)+C$ .

$= \frac{1}{4} sin (u) + \frac{1}{2} x + C$ $=1/4sin(u)+1/2x+C$

Since $u = 2 x$ $u=2x$ :

$= \frac{1}{4} sin (2 x) + \frac{1}{2} x + C$ $=1/4sin(2x)+1/2x+C$

Note that this can be many different ways, since $sin (2 x) = 2 sin x cos x$ $sin(2x)=2sinxcosx$ .

Science

Math

Humanities

... and beyond

What is the integral of ${(cos x)}^{2}$ $(cosx)^2$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

What is the integral of (cosx)2 (cosx)^2?

1 Answer

Explanation:

Related questions

Impact of this question

What is the integral of ${(cos x)}^{2}$ $(cosx)^2$ ?