eTrigIntegral

Example Evaluate the following integral.．

$\displaystyle \int \sin^{3}{x}\cos^{3}{x} dx$

Answer Note that every integral of trig functions can be transformed to the integral of rational functions.

1.	The integration of a sum is the sum of the integration． $\int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx$
2.	The integration of a constant multiple is the constant times the integral． $\int (\alpha f(x))dx = \alpha \int f(x)$
3.	Integration By Substitution． $\int f(x)dx = \int f(\Phi(t))(\Phi(t))'dt$
4.	Integration By Parts. $\int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx$
5.	Partial Fraction． $\frac{3x+2}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$
6.	Basic integration formula． $\int e^{x}dx = e^{x} + c, \int \sin{x} = -\cos{x}+c, \int \frac{1}{x}dx = \log\vert x\vert + c$

In this integrand, we can express $\sin^{3}{x}\cos^{3}{x}$ as

$\displaystyle \sin^{3}{x}\cos^{2}{x} \cos{x} = \sin^{3}{x}(1 - \sin^{2}{x})\cos{x}$

Then by lettint $t = \sin{x}$ , $dt = \cos{x}dx$ , we can express

$\displaystyle \int \sin^{3}{x}\cos^{3}{x} dx= \int \sin^{3}{x}(1 - \sin^{2}{x})\cos{x}dx = \int t^{3}(1-t^2)dt$

Thus, we have

$\displaystyle \int \sin^{3}{x}\cos^{3}{x} dx$	$\displaystyle =$	$\displaystyle \int \sin^{3}{x}(1 - \sin^{2}{x})\cos{x} dx$
	$\displaystyle =$	$\displaystyle \int t^{3}(1 - t^{2})dt = \frac{t^{4}}{4} - \frac{t^{6}}{6} + c$
	$\displaystyle =$	$\displaystyle \frac{\sin^{4}{x}}{4} - \frac{\sin^{6}{x}}{6} + c$