Fundamental Theorem:
The fundamental theorem is essential to integration as it allows us to estimate integrals without using Riemann sums. The theorem says that the integral of a function is its antiderivative. Likewise, if we were to differentiate an integral, we would be left with the original function. For definite integrals, we find the difference between the antiderivative of the upper bound and lower bound. This now allows us to find the area under curves formed by two different lines- which was much more tedious using Riemann sums.

Antiderivatives and Linearity of Integration:
To use the fundamental theorem, we need to be able to find the antiderivative of the original function. To do so, the most basic technique would be to use inspection. Is it a geometric shape that we can calculate the area of? Or we can split the integral into sums of individual integrals and find the antiderivative of each. An easy way is to find some function that when differentiated will result as the function that we’re integrating. The linearity of Integration says we can factor out constants from the integral to simply the function we’re integrating. This is often helpful when the derivative of part of the function is missing a constant or has an unwanted constant.

Substitution:
Often times, it is hard to see how to find the antiderivative of the integral. But by using u-substitution or q-substitution… we can define part of the function within the integral as a variable and differentiate the substitution we used. Then plug the substitution and derivative of the substitution back into the integral. This will usually simplify the integral into a less confusing function. Then we find the antiderivative and substitute back the values we put into our variables so that it’s in terms of our original function. This substitution method allows us to chain rule backwards but much more easily than doing it mentally. The second method of substitution involves trigonometry. We use trig-substitution often when we have a square root as a denominator and a constant as the numerator. By drawing a right triangle, we decide a trig method that will simplify the integral. There are several trig equations that come in handy if memorized.
Integration by parts:
When we can’t integrate through inspection or substitution, we will probably need to integrate by parts. To do so we need to use a different method of substitution in order to integrate. We need to assign the function into parts-hence the name integrate by parts. However, in order to use the formula, we must be able to integrate whatever function we have assigned to dv. For the variable u, we must be able to differentiate the function assigned to it. Then we plug in the substitutions into the formula uv-int(vdu). Often times, we