## viernes, 6 de mayo de 2011

### Integration by Sustitution

Lets start by understanding that this method of integration comes from the rule chain method for example,

$\frac{d[f[g(x)]}{dx}=f'[g(x)]g'(x)$ (1)

lets integrate both sides of equation 1

\begin{align*} \int \frac{d[f[g(x)]}{dx}\; dx&=\int f'[g(x)]g'(x)\;dx \\ f[g(x)]&= \int f'[g(x)]g'(x)\;dx \\ \end{align*}

$\int f'[g(x)]g'(x)\;dx$

we can use the substitution

working with

$u=g(x)$ (2)

then

$du=g'(x)\;dx$

we make the substitution on the integral and get,

$\int f'(u) du$

by the fundamental theorem of calcules we get

$\int f'(u) du=f(u)$

we use equation (2) and get that

$\int f'[g(x)]g'(x)dx=f[g(x)]+c$

Example #1

Solve the integral

$\int (x^2+1)^{20} \; dx$

We choose

$u=x^2+1$

and

$du=2x\;dx$

we change the integral to

$\int u^{20} \; \frac{dx}{2}=\frac{1}{2}\int u^{20}\;du=\frac{1}{2}\frac{1}{21}u^{21}=\frac{1}{42}u^{21}$