Steps for Integration Strategy
1. Simplify the integrand, if possible
This step is vital in the integration process. Several integrals can be taken from impossible or very hard to very easy with a little simplification or manipulation. Just remember basic trig and algebraic identities as these can frequently be utilized to simplify the integral.
We employed this design when we were looking at integrals that are involving trig functions. For instance consider the following integral.
∫ cos2 x dx
This integral cannot be done as is, though simply by reminding the identity,
Cos2 x = ½ (1 + cos (2x))
The integral becomes extremely easy to do.
Note that this instance as well shows that simplification does not essentially mean that we'll write the integrand in a "simpler" form. It just only means that we'll write the integrand into a type that we can deal with and this is frequently longer and/or "messier" as compared to the original integral.
See if a "simple" substitution will work
Look to see if a simple substitution can be employed in place of the often more complicated methods from Calculus II. For instance consider both of the next integrals.
∫ (x /x2 - 1) dx
∫ x√(x2 - 1) dx
The first integral can be completed with partial fractions and the second could be completed with a trig substitution.
Though, both could as well be evaluated by using the substitution u = x2 -1 and the work included in the substitution would be considerably less than the work consist of in either partial fractions or trig substitution.
Thus, all time look for quick, simple substitutions before moving on to the much more complicated Calculus II techniques or methods.
3. Identify the type of integral
Note that any type of integral may fall into more than one of these forms. Due to this fact it's generally best to go all the way by the list and identify all achievable types as one may be easier than the other and it's completely possible that the easier type is listed lower in the list.
a. Is the integrand a rational expression (that is the integrand a polynomial divided by a polynomial)? If so, then partial fractions might work on the integral.
b. Determine Is the integrand a polynomial times a trig function, exponential, or logarithm? If it is like this, then integration by parts may work.
c. Determine is the integrand a product of sines and cosines, secant and tangents, or cosecants and cotangents? If it is like this, then the topics from the second section may work. Similarly, don't forget that some quotients that are consisting of these functions can also be completed by using these techniques.
d. Determine does the integrand involve √b2 x2 + a2, √b2x2 - a2, or √a2 - b2 x2? If it is like this, then a trig substitution might work nicely.
e. Determine does the integrand have roots other than those listed above in it? If it is like this, then the substitution u = n√g (x) might work.
f. Determine does the integrand have a quadratic in it? If it is like this, then completing the square on the quadratic might put it into a form that we can deal with this.
4. Determine can we relate the integral to an integral that we already know how to do?
In other words, can we make use of a substitution or manipulation to write the integrand into a form that does fit into the forms we've looked at previously?
A general example here is the following integral.
∫ cos x√1 + sin2 x dx
This integral does not clearly fit into any of the forms explained previously.
Though, with the substitution u = sin x we can reduce or diminish the integral to the form,
∫√1 + u2 du
which is a trig substitution problem.
5. Do we need to use multiple techniques?
In this step we require to ask ourselves if it is possible that we'll require to makes use of multiple techniques. The instance in the previous part is a good example. By using a substitution didn't permit us to actually do the integral. All it did was put the integral and put it into a type that we could use a different method on.
Don't ever get locked into the thought that an integral will only need one step to completely evaluate it. So many will need more than one step
6. Try once again.
If everything that you have tried to this point doesn't work then go back through the procedure and try again. This time attempt a technique that that you didn't use the first time around.