U Substitution Written by tutor Michael B. Introduction By now, you have seen one or more of the basic rules of integration. These rules are so important and commonly used that many calculus books have these formulas listed on their inside front and/or back covers. Here are a few of them (you may not have learned all of these yet): These rules are so commonly published,... read more

Properties of Integrals Here is a list of properties that can be applied when finding the integral of a function. These properties are mostly derived from the Riemann Sum approach to integration. Additive Properties When integrating a function over two intervals where the upper bound of the first is the same as the first, the integrands can be combined. Integrands can also... read more

Integration by Parts Integration by Parts is a method of integration that transforms products of functions in the integrand into other easily evaluated integrals. The rule is derivated from the product rule method of differentiation. Recalling the product rule, we start with We then integrate both sides We then solve for the integral of f(x)g'(x) Integration... read more

The Fundamental Theorem of Calculus (FOTC) The fundamental theorem of calculus links the relationship between differentiation and integration. We have seen from finding the area that the definite integral of a function can be interpreted as the area under the graph of a function. It justifies our procedure of evaluating an antiderivative at the upper and lower bounds of integration... read more

Solid of Revolution - Finding Volume by Rotation Finding the volume of a solid revolution is a method of calculating the volume of a 3D object formed by a rotated area of a 2D space. Finding the volume is much like finding the area, but with an added component of rotating the area around a line of symmetry - usually the x or y axis. (1) Recall finding the area under a curve. Find... read more

Finding the Area with Integration Finding the area of space from the curve of a function to an axis on the Cartesian plane is a fundamental component in calculus. Definite integration finds the accumulation of quantities, which has become a basic tool in calculus and has numerous applications in science and engineering. While it is used to make formulas in physics more comprehensible,... read more

Integration - Taking the Integral Integration is the algebraic method of finding the integral for a function at any point on the graph. Finding the integral of a function with respect to x means finding the area to the x axis from the curve. The integral is usually called the anti-derivative, because integrating is the reverse process of differentiating. The fundamental theorem... read more

List of Antiderivatives The Fundamental Theorem of Calculus states the relation between differentiation and integration. If we know F(x) is the integral of f(x), then f(x) is the derivative of F(x). Listed are some common derivatives and antiderivatives. Basic Functions Elementary Trigonometric Functions Trigonometric... read more