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…]
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…]
Finding the Integral of a Product of Two Functions
Sometimes the function that you’re trying to integrate is the product of two functions — for example, sin3x and cos x. This would be simple to differentiate with the Product Rule, but integration doesn’t [more…]
How to Integrate a Function Multiplied by a Set of Nested Functions
Sometimes you need to integrate the product of a function (x) and a composition of functions (for example, the function 3x2 + 7 nested inside a square root function). If you were differentiating, you could [more…]
How to Integrate Compositions of Functions
Compositions of functions — that is, one function nested inside another — are of the form f(g(x)). You can integrate them by substituting u = g(x) when [more…]
Substituting with Expressions of the Form f(x) Multiplied by g(x)
When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by g(x). Variable substitution helps to fill the gaps left by the absence of a Product Rule [more…]
Substituting with Expressions of the Form f(x) Multiplied by h(g(x))
When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by h(g(x)), provided that h is a function that you already know how to integrate. [more…]
Using Variable Substitution to Evaluate Definite Integrals
When using variable substitution to evaluate a definite integral, you can save yourself some trouble at the end of the problem. Specifically, you can leave the solution in terms of [more…]
Knowing When to Integrate by Parts
It’s important to recognize when integrating by parts is useful. To start off, here are two important cases when integration by parts is definitely the way to go: [more…]
How to Integrate Odd Powers of Sines and Cosines
You can integrate any function of the form sinm x cosn xwhen m is odd, for any real value of n. For this procedure, keep in mind the handy trig identity sin [more…]
How to Integrate Even Powers of Sines and Cosines
You can integrate even powers of sines and cosines. For example, if you wanted to integrate sin2 x and cos2 x, you would use these two half-angle trigonometry identities: [more…]
Using Identities to Express a Trigonometry Function as a Pair of Functions
You can express every product of powers of trig functions, no matter how weird, as the product of any pair of trig functions. The three most useful pairings are sine and cosine, tangent and secant, and [more…]
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
In many cases, you can untangle hairy rational expressions and integrate them using the anti-differentiation rules plus the Sum Rule, Constant Multiple Rule, and Power Rule. [more…]
