How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Cosine
When you integrate a trig integral that includes cosine, and if the power of cosine is odd and positive, you can convert and then use substitution to integrate. To make this conversion, you need to know [more…]
How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent
Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity: [more…]
How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant
Here’s how you integrate a trig integral that contains tangents and secants where the secant power is even and positive. Like with all tangent/secant integrals, you use the tangent-secant version of the [more…]
How to Integrate Problems with an Even, Positive Power of Tangent
Here’s how you integrate a trig integral that contains tangents (and no secant factors) where the tangent power is even and positive. [more…]
How to Use Trig Substitution to Integrate
With the trigonometric substitution method, you can do integrals containing radicals of the following forms (given a is a constant and u is an expression containing [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 Find a Function’s Average Value with the Mean Value Theorem for Integrals
You can find the average value of a function over a closed interval by using the mean value theorem for integrals. The best way to understand the mean value theorem is with a diagram — check it out below [more…]
How to Find the Volume of a Complicated Shape with the Meat-Slicer Method of Integration
In geometry, you learned how to figure the volumes of simple solids like boxes, cylinders, and spheres. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes [more…]
How to Find the Volume of a Circular Shape with the Stack-of-Pancakes Method of Integration
Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. The stack-of-pancakes technique works [more…]
How to Find the Volume of a Shape Using the Washer Method of Integration
Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. If you have a circular shape with [more…]
How to Find the Volume of a Cylindrical Shape with the Nested-Russian-Dolls Method of Integration
Integration enables you to calculate the volumes of an endless variety of complicated shapes that you can’t handle with regular geometry. You can cut up a solid into thin concentric cylinders and then [more…]
How to Calculate Arc Length with Integration
When you use integration to calculate arc length, what you’re doing (sort of) is dividing a length of curve into infinitesimally small sections, figuring the length of each small section, and then adding [more…]
How to Make Unacceptable Forms Acceptable before Using L'Hôpital’s Rule
You can use L'Hôpital’s rule to find a limit when other methods don’t work. In fact, even if some other method does work, L'Hôpital’s rule is often a good shortcut. If substitution of the limit number [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 Do a Related Rate Problem Involving a Moving Baseball
You can use calculus to determine a rate that’s related to the speed of a moving object. For example, say a pitcher delivers a fastball, which the batter pops up — it goes straight up above home plate. [more…]
How to Determine Limits of Sequences with L'Hôpital's Rule
You can use L’Hôpital’s rule to find limits of sequences. L'Hôpital's rule is a great shortcut for when you do limit problems. Here it is: [more…]
How to Use the nth Term Test to Determine Whether a Series Converges
If the individual terms of a series (in other words, the terms of the series’ underlying sequence) do not converge to zero, then the series must diverge. This is the [more…]
How to Work with Geometric Series
Geometric series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A geometric series is a series of the [more…]
How to Analyze a p-Series
So-called p-series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A [more…]
How to Analyze a Telescoping Series
You don’t see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Consider the following series: [more…]
Using the Direct Comparison Test to Determine If a Series Converges
The direct comparison test is a simple, common-sense rule: If you’ve got a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is larger than [more…]
How to Use the Limit Comparison Test to Determine Whether a Series Converges
The idea behind the limit comparison test is that if you take a known convergent series and multiply each of its terms by some number, then that new series also converges. And it doesn’t matter whether [more…]
