User:Able806/integration

Methods of Integration

Integration by Parts

Integration by parts is a special technique to facilitate the integration of the product of two functions that otherwise lack an obvious integral. This technique utilizes the insight of the product rule.

The rule for integration by parts is stated as follows:

$\int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx,$

This rule is often useful when one function is a power of x and the other function is a trigonometric function or e raised to a power of x.

Note that it may be necessary to repeat the integration by parts several times, one for each power of x.

Partial Fractions

Integration by partial fractions is a technique to facilitate the integration of a rational expression by partial fraction decomposition.

Given an integral

$\int {\frac {3x+11}{x^{2}-x-6}}dx$

The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this gives,

${\frac {3x+11}{(x-3)(x+2)}}={\frac {A}{x-3}}+{\frac {B}{x+2}}$

This allows us to split the fraction in to sums by cross multiplying the denominators,

${\frac {3x+11}{(x-3)(x+2)}}={\frac {A(x+2)+B(x-3)}{(x-3)(x+2)}}$

Therefore we can restate the problem,

$3x+11=A(x+2)+B(x-3)$

Now we can solve for A and B by subsituting x with a value that allows the term to go to 0. For example,

We let :$x=-2$,

$3(-2)+11=A(-2+2)+B(-2-3)$

$5=A(0)+B(5)$

$B=-1$

We let :$x=3$,

$3(3)+11=A(3+2)+B(3-3)$

$20=A(5)+B(0)$

$A=4$

We then plug in the values of A and B and get,

${\frac {4}{x-3}}-{\frac {1}{x+2}}$

Now we can solve the integral.

$\int {\frac {3x+11}{x^{2}-x-6}}dx=\int {\frac {4}{x-3}}-{\frac {1}{x+2}}dx$

$\int {\frac {3x+11}{x^{2}-x-6}}dx=\int {\frac {4}{x-3}}dx-\int {\frac {1}{x+2}}dx$

$\int {\frac {3x+11}{x^{2}-x-6}}dx=4ln|x-3|-ln|x+2|+c$