## Review of Integration Techniques

This page contains a review of some of the major techniques of
integration, including

and a table of common integrals. A more thorough
and
complete treatment of these methods can be found in your textbook (or any
general calculus book). There is also a
page of practice problems with answers which
might be helpful.

**Substitution**

In some cases, an integral can be altered into a manageable form by
just changing variables. If the integrand can be written in the form f(g(x))g'(x)
then we may make the substitution u=g(x) (which implies du=g'(x)dx) and
integrate as follows:

If the function f(u) has an easily indentifiable antiderivative then all
is well. If not, another substitution or integration method may be needed.
The common choices for g(x) are arguments of trigonometric functions and
functions raised to powers.

#### Example

To evaluate the integral

we may consider choosing

**u = 3 sin**^{2}(x) + e^{8}

du = 6 sin(x)
cos(x) dx

Then

where C is a constant of integration.

**Integration by Parts**

Let u(x) and v(x) be two differentiable
functions.
An easy way to get the
formula for integration by parts is as
follows:

In the case of a definite integral we have

Integration by parts is useful in "eliminating" a part of the integral
that makes the integral difficult to do. The annoying part of the integral
is often chosen to be u(x).

#### Example

For the function

we notice that this function could be integrated with a substitution if
the x^3 term
were only an x. This is the perfect scenario for integration by parts. If
we choose u(x)=x^2 and v'(x)=xe^(x^2), then we have

which implies

where C is a constant of integration.

**Integrating Rational Functions**

A rational function is a function that can be expressed as the ratio of
two polynomials. Consider integrating the rational function

To integrate such a function we use the
method
of partial fractions to split the fraction into easily integrable pieces:

Now the integral is easy:

This method is designed for fractions with a polynomial of lesser degree
in the numerator than in the denominator. If this is not the case, long (or
synthetic) division must be carried out first and then the method of partial
fractions can be used on the remainder term (if necessary).

Below is a table of common integrals. Other integrals can be found
in your textbook, a table of integrals and series, or any decent calculus
book.

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of
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