# Problem of the Week

## Updated at Sep 2, 2013 4:27 PM

How can we find the integral of $$\cos^{2}x$$?

Below is the solution.

$\int \cos^{2}x \, dx$

 1 Use Pythagorean Identities: $$\cos^{2}x=\frac{1}{2}+\frac{\cos{2x}}{2}$$.$\int \frac{1}{2}+\frac{\cos{2x}}{2} \, dx$2 Use Sum Rule: $$\int f(x)+g(x) \, dx=\int f(x) \, dx+\int g(x) \, dx$$.$\int \frac{1}{2} \, dx+\int \frac{\cos{2x}}{2} \, dx$3 Use this rule: $$\int a \, dx=ax+C$$.$\frac{x}{2}+\int \frac{\cos{2x}}{2} \, dx$4 Use Constant Factor Rule: $$\int cf(x) \, dx=c\int f(x) \, dx$$.$\frac{x}{2}+\frac{1}{2}\int \cos{2x} \, dx$5 Use Integration by Substitution on $$\int \cos{2x} \, dx$$.Let $$u=2x$$, $$du=2 \, dx$$, then $$dx=\frac{1}{2} \, du$$6 Using $$u$$ and $$du$$ above, rewrite $$\int \cos{2x} \, dx$$.$\int \frac{\cos{u}}{2} \, du$7 Use Constant Factor Rule: $$\int cf(x) \, dx=c\int f(x) \, dx$$.$\frac{1}{2}\int \cos{u} \, du$8 Use Trigonometric Integration: the integral of $$\cos{u}$$ is $$\sin{u}$$.$\frac{\sin{u}}{2}$9 Substitute $$u=2x$$ back into the original integral.$\frac{\sin{2x}}{2}$10 Rewrite the integral with the completed substitution.$\frac{x}{2}+\frac{\sin{2x}}{4}$11 Add constant.$\frac{x}{2}+\frac{\sin{2x}}{4}+C$Donex/2+sin(2*x)/4+C