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Unit: Integration for physics (Prerequisite)

Antiderivatives

If f' is the derivative of f, then f is the antiderivative of f'. To find the antiderivative of a function we need to perform some kind of reverse differentiation. Learn about it here.

Indefinite integrals of common functions

Indefinite integrals (or antiderivatives) are really just backward differentiation. Therefore, the indefinite integral of eˣ is eˣ+c, the indefinite integral of 1/x is ln(x)+c, the indefinite integral of sin(x) is -cos(x)+c, and the indefinite integral of cos(x) is sin(x)+c.

Definite integral as area

Introducing the definite integral. It's basically a way to represent the area under a given curve with left-hand and right-hand bounds.

Riemann sums

Riemann sums is the name of a family of methods we can use to approximate the area under a curve. Learn about the different ways and how they are constructed.

Definite integral evaluation

The fundamental theorem tells us how we can evaluate definite integrals. Now let us put this to use and evaluate some!

u-substitution

u-substitution is an extremely useful technique. Harnessing the power of the chain rule, it allows us to define a new variable (common denoted by the letter u) as a function of x, and obtain a new expression which is (hopefully) easier to integrate.

Area & net change

The area under a rate function gives the net change. This result of the fundamental theorem of calculus is being put here to use with some real-world problems.