Calculus

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How to Use Math Root Rules

When using math root rules, first note that you can’t have a negative number under a square root or any other evennumber root — at least, not in basic calculus. Here are a couple of easy rules to begin [more…]

How to Solve a Quadratic Equation by Factoring

A quadratic equation is any second degree polynomial equation — that’s when the highest power of x, or whatever other variable is used, is 2. You can solve quadratic equations by factoring. [more…]

How to Find Local Extrema with the Second Derivative Test

The Second Derivative Test is based on two prize-winning ideas: First, that at the crest of a hill, a road has a hump shape — in other words, it’s curving down or concave down. And second, at the bottom [more…]

Solving Differential Equations Using an Integrating Factor

A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an [more…]

How to Antidifferentiate Any Polynomial Using the Sum, Constant Multiple, and Power Rules

The anti-differentiation rules for integrating greatly limit how many integrals you can compute easily. In many cases, however, you can integrate any polynomial in three steps by using the Sum Rule, Constant [more…]

How to Use Identities to Integrate Trigonometry Functions

You’ll be surprised how much headway you can often make when you integrate an unfamiliar trigonometry function by first tweaking it using the Basic Five trig identities: [more…]

Computing Integrals and Representing Integrals as Functions

In trying to understand what makes a function integrable, you first need to understand two related issues: difficulties in computing integrals and representing integrals as functions. [more…]

Understanding What Makes a Function Integrable

When mathematicians discuss whether a function is integrable, they aren’t talking about the difficulty of computing that integral — or even whether a method has been discovered. Each year, mathematicians [more…]

Use a Shortcut for Integrating Compositions of Functions

You can use a shortcut to integrate compositions of functions — that is, nested functions of the form f(g(x)). Technically, you’re using the variable substitution [more…]

Using the Product Rule to Integrate the Product of Two Functions

The Product Rule enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that’s useful for integrating [more…]

How to Integrate Even Powers of Secants with Tangents

You can integrate even powers of secants with tangents. If you wanted to integrate tanm xsecn x when n is even — for example, tan8 x sec6 x — you would follow these steps: [more…]

How to Integrate Odd Powers of Tangents with Secants

You can integrate odd powers of tangents with secants. To integrate tanm x secnx when m is odd — for example, tan7x sec9 x — you would follow these steps: [more…]

Using Trigonometry Substitution to Integrate a Function

Trig substitution allows you to integrate a whole slew of functions that you can’t integrate otherwise. These functions have a special, uniquely scary look about them and are variations on these three [more…]

Integrate a Function Using the Secant Case

When the function that you’re integrating includes a term of the form (bx2 a2)n, draw your trig substitution triangle for the secant case. For example, suppose that you want to evaluate this integral [more…]

Knowing When to Avoid Trigonometry Substitution

It’s useful to know when you should avoid using trig substitution. With some integrals, it’s better to expand the problem into a polynomial. For example, look at the following integral: [more…]

Setting Up Partial Fractions When You Have Distinct Linear Factors

Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. The simplest case in which partial fractions are helpful [more…]

How to Distinguish Proper and Improper Rational Expressions

Integration by partial fractions works only with proper rational expressions, but not with improper rational expressions. Telling a proper fraction from an improper one is easy: A fraction a/b is proper [more…]

How to Split One Definite Integral into Two Definite Integrals

When solving area problems, you sometimes need to split an integral into two separate definite integrals. Here’s a simple but handy rule for doing this that looks complicated but is really very easy: [more…]

How to Evaluate an Improper Integral that Is Horizontally Infinite

Improper integrals are useful for solving a variety of problems. A horizontally infinite improper integral contains either ∞ or –∞ (or both) as a limit of integration. [more…]

Find the Area Between Two Functions

To find an area between two functions, you need to set up an equation with a combination of definite integrals of both functions. For example, suppose that you want to calculate the shaded area between [more…]

Using the Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals guarantees that for every definite integral, a rectangle with the same area and width exists. Moreover, if you superimpose this rectangle on the definite integral, [more…]

Finding the Volume of a Solid with Congruent Cross Sections

Knowing how volume is measured without calculus pays off big-time when you step into the calculus arena. This is strictly no-brainer stuff — some basic, solid geometry that you probably know already. [more…]

Finding the Volume of a Solid with Similar Cross Sections

Finding the volume of a prism or cylinder is pretty straightforward. But what if you need to find the volume of a shape like the one shown here? In this case, slicing parallel to the base always results [more…]

How to Measure the Volume of an Irregular-Shaped Solid

You can measure the volume of any irregular-shaped solid with a cross section that’s a function of x. In some cases, these solids are harder to describe than they are to measure. For example, have a look [more…]

How to Measure the Volume of an Object by Turning It on Its Side

Sometimes if you want to measure the volume of an object, you need to turn it on its side so that you can use the meat-slicer method. This method works best with solids that have similar cross sections [more…]

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