The definite integral is defined as the area "under" a function in a specified interval. There is no systematic set of rules that allow us to compute integrals of all functions we can write down the way we can compute derivatives.
We can, however, compute integrals of many functions. We do so by exploiting the rules for differentiation and the fact that taking the integral is the inverse to the operation of differentiation.
The linearity of integration allows us to obtain the integral of a sum as the sum of the integral of its summands, and the integral of a constant multiple of an integrand as that same multiple of the integral.
The power rule for differentiation along with the fact that dlnx=dx/x, allows us to integrate any power and hence, any polynomial or weighted sum of positive and negative powers.
A trick, called partial fraction expansion allows us to integrate any rational function of the variable of integration if we can factor out its denominator into linear or quadratic factors.
The chain rule for differentiation translates into a rule for changing variable of integration. By making a change of variable in an integral we know how to do, we can make it into something new that we can therefore also know how to do.
We can integrate any sum of exponents, that is, of terms each of the form cekxdx, by using the rule for differentiating exponents changing variable of integration from x to kx.
We can integrate any combination of the form sinmxcosnx for positive integer m and n by making trigonometric substitutions and using trigonometric identities.
We can use inverse trigonometric substitutions to use these results to evaluate certain integrals with factors like ; with further substitutions and completing the square we can do similar integrals with a factor of the form .
Integration by parts is the integration trick corresponding to the product rule for differentiation. It can be used to perform integrals of polynomials times exponents, to do integrals of the form sinmxcosnx for non integer n and/or m, and to do many integrals involving logarithms.
You should attempt to develop a list of integrals and types of integrals that you can recognise as evaluatable by one or another of these methods.
You should also practice each of them so that, if trapped on a desert island or on an examination withonly a pencil and paper, you could recognise when to apply it and actually apply it.
There are computer programs which are experts at this. They will tell you the formula for any evaluatable integral you enter to it (and give a numerical evaluation of any others.) This means that all is not lost if you are a poor integrator. (In the same way, there are computer programs which can read written material aloud to you; which means that all is not lost if you are a poor reader).
Direct and inverse substitution:
Given an integral, , a direct substitution is one in which you pick a function u(x), form du(x) and reexpress as a function of u.
Then you have
An inverse substitution is one obtained by letting x be some function of the new variable u.
In this case, you must solve for u in terms of x at the end, anyway. Of course the substitutions u = lnx and x = eu are identical and lead to the same result.
Direct trigonometric substitutions are used to make integrals involving trigonometric functions into integrals of rational functions or functions containing factors like as in section 2 here. Inverse trigonometric substitution is used to provide evaluations for integrals containing factors like as described in section 3.