Integration by Substitution
So the integral reduces to:
Integration by Parts
Integration by parts can be attempted with any integral whatsoever. The best way to try is simply to guess and check, since applying the rules methodically can be exhausting sometimes. The way to get good is to begin by practicing the product rule:
So take any integral whatsoever, like:
Integration by Partial Fractions
Becoming efficient at decomposing fractions as partial fractions is a great skill to have for any scientist. Being able to decompose simple fractions mentally allows for a mental flexibility that is useful when looking at complicated formulas.
we can of course combine these as:
Complicated trigonometric integrals appear in many areas of physics and engineering. One area where they are found very commonly is in experimental particle physics during calculations of cross-sections. Essentially the idea is to find the number of particles that will bounce into a certain direction given that they undergo a collision somewhere. Since the detectors are usually spherical, there are often a number of trigonometric factors that must be integrated over simultaneously.
Another important application for trigonometric integrals is in the area of frequency analysis which is used by sound engineers, physicists, and even by business theorists. In this theory a signal is broken into its individual frequencies. This involves a topic called Fourier analysis, which rests upon integration over many powers of sines and cosines.
Trigonometric substitutions should be thought of as tricks that can be very useful for solving a variety of integrals. The only way to recognize when a trigonometric substitution should be applied is through experience. On the other hand, these tricks must be applied to a variety of integrals if they are to be solved.
Often the underlying cause for needing a trig substitution is that the function is best described in a different coordinate system. The only coordinates that have been discussed in this text are Cartestian or Rectangular coordinates. However, Polar Coordinates are also very useful in many contexts. These are described by the following diagram:
Integrals with infinite limits or that pass over points of discontinuity are extremely important in physics as well as engineering. It is actually kind of miraculous that the area under a curve like the one below is not infinite:
This area is not infinite even though the tail of the graph goes off to the right all the way out the infinity and is never zero. Similarly, a graph can go off to infinity in the vertical direction but still have an area that is not infinite. This is part of what is nice about having numbers. The picture looks like the area is infinity, but the math gives us a different answer that we can use.
Ordinary Differential Equations
How does calculus appear in actual physics or engineering applications? The answer is through differential equations. Basically a theory tells us what the differential equation should look like, and then to find out what will happen we need to either solve the equation or else numerically approximate a solution based on the starting conditions. Often the result depends very delicately upon the starting conditions, and when the result is drastically different for different starting conditions we have chaos.
Most of the time scientists will be able to look at a differential equation and understand some basic ideas about how the function behaves. For example, suppose we think abstractly about the simple differential equation:
The different possible solutions (each of the lines above) simply represent different possible values for the constant.
Now consider the slightly more complicated differential equation: