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### Find the Area Under More Than One Function

Sometimes, a single geometric area is described by more than one function. For example, suppose that you want to find the shaded area shown in the following figure, the area under [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, sin^{3}*x* 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 3*x*^{2} + 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 sin^{m}*x* cos^{n}*x*when *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 sin^{2} *x* and cos^{2} *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…]

### When to Use Variable Substitution with Integrals

Variable substitution comes in handy for some integrals. The anti-differentiation formulas plus the Sum Rule, Constant Multiple Rule, and Power Rule allow you to integrate a variety of common functions [more…]

### Find the Integral of Nested Functions

Sometimes you need to integrate a function that is the composition of two functions — for example, the function 2*x* nested inside a sine function. If you were differentiating, you could use the Chain Rule [more…]

### Integrate a Function Using the Sine Case

When the function you’re integrating includes a term of the form (*a*^{2} – *bx*^{2})* ^{n}*, draw your trig substitution triangle for the

*sine case.*For example, suppose that you want to evaluate the following integral [more…]

### Integrate a Function Using the Tangent Case

When the function you’re integrating includes a term of the form (*a*^{2} + *x*^{2})* ^{n}*, draw your trigonometry substitution triangle for the tangent case

*.*For example, suppose that you want to evaluate the following [more…]

### Setting Up Partial Fractions When You Have Distinct 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. One case where you can use partial fractions is when the [more…]

### Setting Up Partial Fractions When You Have Repeated 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. One case where you can use partial fractions is with repeated [more…]

### How to Evaluate an Improper Integral that Is Vertically Infinite

Improper integrals are useful for solving a variety of problems. A *vertically infinite* improper integral contains at least one vertical asymptote. Vertically infinite improper integrals are harder to recognize [more…]

### How to Find the Volume of a Solid between Two Surfaces of Revolution

If you want to find the volume of a solid that falls between two different surfaces of revolution, you can use the meat-slicer method to do this. The *meat-slicer method* [more…]

### Finding the Area of a Surface of Revolution

The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.

To find the area of a surface of revolution between [more…]

### Using the Shell Method to Find the Volume of a Solid of Revolution

The shell method is useful when you’re measuring a volume of revolution around the *y*-axis. For example, suppose that you want to measure the volume of the solid shown in this figure. [more…]

### Comparing Converging and Diverging Sequences

Every infinite sequence is either convergent or divergent. A *convergent* *sequence* has a limit — that is, it approaches a real number. A *divergent* *sequence* [more…]

### Using the Constant Multiple Rule for Simplifying a Series

The Constant Multiple Rule for integration allows you to simplify an integral by factoring out a constant. This option is also available when you’re working with series. Here’s the rule: [more…]

### Using the Sum Rule for Simplifying a Series

The Sum Rule for integration allows you to split a sum inside an integral into the sum of two separate integrals. Similarly, you can break a sum inside a series into the sum of two separate series: [more…]