Integration

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How to Do Simple Integration by Parts

Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]

How to Do Integration by Parts

Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]

How to Use Sigma Notation

For adding up long series of numbers like the rectangle areas in a left, right, or midpoint sum, sigma notation comes in handy. Here’s how it works. Say you wanted to add up the first 100 multiples of [more…]

How to Do Integration by Parts More than Once

Sometimes you have to use the integration-by-parts method more than once because the first run through the method takes you only part way to the answer. [more…]

How to Find Antiderivatives by Guessing and Checking

The guess-and-check method works when the integrand— that’s the thing you want to antidifferentiate (the expression after the integral symbol, not counting the [more…]

How to Find Antiderivatives with the Substitution Method

When a function’s argument (that’s the function’s input) is more complicated than something like 3x + 2 (a linear function of x — that is, a function where [more…]

How to Write Riemann Sums with Sigma Notation

You can use sigma notation to write out the right-rectangle sum for a function. For example, say you’ve got f(x) = x2 + 1.

By the way, you don’t need sigma notation for the math that follows. It’s just [more…]

How to Integrate by Using Partial Fractions when the Denominator Contains Only Linear Factors

You can use the partial fractions method to integrate rational functions (Recall that a rational function is one polynomial divided by another.) The basic idea behind the partial fraction approach is “unadding” [more…]

Integrating Using Partial Fractions when the Denominator Contains Irreducible Quadratic Factors

You can use the partial fractions method to integrate rational functions, including functions with denominators that contain irreducible quadratic factors [more…]

How to Solve Improper Integrals for Functions that Have Vertical Asymptotes

You solve improper integrals by turning them into limit problems. You can’t just do them the regular way. Here’s how you solve improper integrals for functions that have vertical asymptotes. There are [more…]

How to Solve Improper Integrals that Have One or Two Infinite Limits of Integration

One of the ways in which definite integrals can be improper is when one or both of the limits of integration are infinite. You solve this type of improper integral by turning it into a limit problem where [more…]

How to Find the Volume and Surface Area of Gabriel's Horn

Finding the volume and surface area of this horn problem may blow your mind. Gabriel’s horn is the solid generated by revolving about the x-axis the unbounded region between [more…]

How to Find Area with the Shortcut Version of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus has a shortcut version that makes finding the area under a curve a snap. Here it is. Let F be any antiderivative of the function [more…]

Integration by Parts Problems where You Go around in Circles

Sometimes if you use integration by parts twice, you get back to where you started from — which, unlike getting lost, is not a waste of time. [more…]

The Riemann Sum Formula For the Definite Integral

The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows [more…]

The Sum Rule, the Constant Multiple Rule, and the Power Rule for Integration

When you perform integration, there are three important rules that you need to know: the Sum Rule, the Constant Multiple Rule, and the Power Rule.

The Sum Rule for Integration tells you that it’s okay to [more…]

Integration by Parts with the DI-agonal Method

The DI-agonal method is basically integration by parts with a chart that helps you organize information. This method is especially useful when you need to integrate by parts more than once to solve a problem [more…]

How to Use Integration by Parts

When doing Calculus, the formula for integration by parts gives you the option to break down the product of two functions to its factors and integrate it in an altered form. To use integration by parts [more…]

Calculus: Techniques of Integration

You'll find that there are many ways to solve an integration problem in calculus. The following list contains some handy points to remember when using different integration techniques: [more…]

Solving Integration Problems in Calculus

Calculus riddle: What do the Mean Value Theorem, the Washer and Shell Methods, and the Arc Length and Surface of Revolution formulas have in common? They all involve integration. Integration is very fancy [more…]

How to Do an Area Approximation Using Sigma Notation

Sigma notation comes in handy when you’re approximating the area under a curve. For example, express an 8-right-rectangle approximation of the area under [more…]

Quickly Compute Definite Integrals Using the Fundamental Theorem

Here is a super-duper shortcut integration theorem that you'll use for the rest of your natural born days — or at least till the end of your stint with calculus. This shortcut method is all you need for [more…]

How to Find Antiderivatives Using Reverse Rules

You can use reverse rules to find antiderivatives. The easiest antiderivative rules are the ones that are the reverse of derivative rules you already know. These are automatic, one-step antiderivatives [more…]

How to Solve Integrals Using Integration by Parts

You can think of integrating by parts as the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]

Integrate When the Powers of Sine, Cosine Are Even, Nonnegative

When the powers of both sine and cosine are even and nonnegative, you can convert the integrand into odd powers of cosines by using the following trig identities. [more…]

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