30 - Grand Review - Integral Calculus

An integral is a mathematical function that attempts to compute the area under the curve of a function.
In equivalent terms, the integral represents the area between the curve and the x-axis.

If differentiating gives the derivative, integration undoes that, giving the antiderivative/primitive.
In essence, integration is the opposite of differentiation.

If the derivative represents how much the function is instantaneously changing at a point, the integral represents how much that function has changed, or, how much change it has accumulated across that interval of points

If you have an indefinite integral, ALWAYS ADD THE +c AT THE END!!!
An indefinite integral does not give one specific function, but an infinite family of the same function that differ by a constant.

A definite integral is an integral that is taken between two bounds. If an indefinite integral would result in a function that represents the area under the curve for an x-value, the definite integral then evaluates that exact function between those points.

• You can split an integral of a sum of functions into a sum of integrals.
  
• You can move a constant multiple in and out of the integral.

An integral that starts and ends at the same point will be zero, because you can't have a surface for a line or point.

WARNING: THIS DOES NOT APPLY TO n=-1 !!!

Much like the Lagrange MVT for derivatives, integrals also have a mean value across an interval.

We know that any differentiable function has derivatives that are continuous. In a sense, all functions that are continuous are a derivative of some other function. This means that all we need to prove integrability is continuity.

This technique can be used when you have an integral with a function and its derivative in the same equation. You find u, u', then replace u'dx with du, integrating for du. When finished, you substitute again for x.
In essence, integration by u-substitution undoes the chain rule.

NOTE THAT IF THIS INTEGRAL IS DEFINITE, THE STARTING BOUNDS WILL BE IN TERMS OF X, SO WHEN U-SUBSTITUTING YOU WILL HAVE TO RE-WRITE THESE BOUNDS FOR U IN TERMS OF X!

This technique is used when you have an integral of a product of functions, and one of them is some derivative of some random function. This, in essence, undoes the product rule of a derivative.

Whenever you have a fraction of polynomials, where the denominator is a product of two monomials or polynomials of x, you can "disassemble" or "expand" that fraction into a sum of simpler fractions, splitting the integral and making it easier or possible to integrate.

Whenever you have some form of you can substitute for , calculating dx as the derivative of multiplied by , then integrating the whole expression in terms of theta. You then substitute the final result for x again.

Whenever you have some form of you can substitute for , calculating dx as the derivative of multiplied by , then integrating the whole expression in terms of theta. You then substitute the final result for x again.

Knowing that the integral represents the area under the curve of a function, an improper integral takes the area of an infinitely large curve or approaches a point where the function has a vertical tangent. This is done instead as the limit of a definite integral where this infinite boundary point is replaced by a variable for which the limit will go to infinity.

Similarly,