Let’s get on with the strategy.
2. See if a simple substitution will work
Look to see if a simple substitution can be used instead of the often more complicated methods. For example consider both if the following integrals.
For examples:
c) Manipulate the integrand
d) Relate the integral to an integral that we already know how to do.
When you have build up some experience in integration, you may be able to use a method on a given integral that is similar to a method you have already used on a previous integral. Or you may be able to express the given integral in terms of a previous one.
For example:
and if
has previously been evaluated, then that calculation can be used in the present problem.
e) Use several methods.
Sometimes 2 or 3 methods are required to evaluate an integral. The evaluation could involve several successive substitutions of different types, or it might combine integration by parts with one or more substitutions.
1. Simplify the integrand, if possible
Many integrals can be taken from impossible or very difficult to very easy with a little simplification or manipulation. Don't forger basic trigonometric and algebraic identities as these can often be used to simplify the integral. For example consider the following integral;
Look to see if a simple substitution can be used instead of the often more complicated methods. For example consider both if the following integrals.
The first integral can be done with partial fractions and the second could be done with a trig substitution. However, both could also be evaluated using the substitution 
.
.
and the work involved in the substitution would be significantly less than the work involved in either partial fractions or trigonometric substitution.
So, always look for quick, simple substitutions before moving on to the more complicated techniques.
3. Identify the form of integral
If steps 1 and 2 have not led to the solution, then we take a look at the form of the integrand f(x).
Trigonometric functions
Rational functions
f (x) is P(x) divided by D(x)
[use partial fractions]
[use partial fractions]
LPET
f (x) is a product of polynomial, trigonometric function, exponential, or logarithm [use integration by parts]
f (x) involve 
[use trigonometric substitutions]
[use trigonometric substitutions]
Quadratic form
[completing the square on the quadratic]
[completing the square on the quadratic]
4. Try again
If the first 3 steps have not produced the answer, remember that there are basically only 2 methods of integration : substitution and by parts.
a) Try substitution
Even if no substitution is obvious, some inspiration or ingenuity may suggest an appropriate substitution.
b) Try by parts
Although integration by parts is used most of the time on LPET, it is sometimes effective on single functions.
For examples:
c) Manipulate the integrand
Algebraic manipulations (rationalizing the denominator/ using trigo identities) may be useful in transforming the integral into an easier form. For examples:
d) Relate the integral to an integral that we already know how to do.
When you have build up some experience in integration, you may be able to use a method on a given integral that is similar to a method you have already used on a previous integral. Or you may be able to express the given integral in terms of a previous one.
For example:
and if
has previously been evaluated, then that calculation can be used in the present problem.e) Use several methods.
Sometimes 2 or 3 methods are required to evaluate an integral. The evaluation could involve several successive substitutions of different types, or it might combine integration by parts with one or more substitutions.
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