Derivatives find instantaneous rates of change. By contrast, integrals find areas under curves.
The integral symbol ∫ is derived from the word “sum.” To estimate the area under the curve, we can approximate the curve using rectangles and compute the sum of their area, as pictured below.
This is called a Riemann sum. Dr. John Urschel explains Riemann sum in this lesson clip from his Outlier course:
However, this sum either overestimates or underestimates the area under the curve.
To compute the precise area under the curve, we take the limit of the Riemann sum as the number of subdivisions approaches infinity. We can think of this limit as the sum of infinitely many rectangles, where each rectangle’s height is y, and each rectangle’s width is a tiny change in x. This gives us the definition of a definite integral, where A represents the area under the curve on [a,b].
What does ∫ mean? The symbol ∫ is called the integral sign. The inside function f(x) is called the integrand.
To evaluate a definite integral, we take the difference between the indefinite integral of the function evaluated at a and the indefinite integral of the function evaluated at b.
An indefinite integral finds the antiderivatives of a function, which we usually denote by F(x):
If F’(x)=f(x), then F(x) is an antiderivative of f(x). This means that taking the derivative of F(x) gives back f(x). Evaluating an indefinite integral answers the question, “What functions, when differentiated, give back the integrand we started with?”
You can think about this process as differentiation in reverse. If ∫f(x)dx=F(x)+C, then F’(x)=f(x). As mentioned earlier, the antiderivative function F(x) can be used to calculate all possible areas under the curve by taking the difference of F(x) evaluated at the endpoints of an interval.
It might be helpful to think about this concept another way, too. If integrating a derivative of a function, then we have:
C represents the constant of integration. Since the derivative of any constant is zero, C can take any value. Thus, the indefinite integral gives us a family of antiderivative functions.
Top 6 Integral Shortcuts
How do you integrate quickly? We’ll cover 6 handy tricks for evaluating integrals.
1. Memorize Basic Integral Formulas
The most practical integral trick is to memorize the most common integral examples. Over time, these examples have become standard integral equations. These basic integral formulas are essential tools for evaluating integrals.
5 Basic Integration Formulas
These equations are given below.
Constant Multiplier Rule
∫kf(x)dx=k∫f(x)dx for some constant k
∫xndx=n+1xn+1+C for some real number n
∫adx=ax+C for some constant a
Dr. Tim Chartier, an Outlier instructor of Calculus, explains several of these rules:
4 Integral Formulas for Specific Function Types
Once you’re familiar with the basic rules above, you can memorize these integral formulas for specific function types:
For example, what is the integral shortcut for ∫xdx? To determine this, we can use the power rule. In this case, our exponent is n=1. Using the power rule, we have ∫x1dx=1+1x1+1=2x2+C.
Similarly, how do you find the integral of x2? Again, we can use the power rule. In this case, n=2. Using the power rule, we have ∫x2dx=2+1x2+1=3x3+C.
What is the integral of sin(x) dx? For this integral, we can use the trigonometric rules. The sine rule tells us that ∫sin(x)dx=−cosx+C.
2. Use LaTeX to Streamline Formatting
How do you type an integral on a keyboard? The easiest option is to use LaTeX, which is a typesetting system that is optimized to display technical content. You can download LaTeX onto your computer, or you can make an account on a free online LaTeX editor.
To display a definite integral using LaTeX, you can use the code “∫f(x)dx=F(x)+C.” The dollar signs indicate math mode in LaTeX. To display an indefinite integral, you can use the code “∫abf(x)dx=F(x)∣∣ab.”
Alternatively, the keyboard shortcuts for Mac and Windows are Option + B and Alt + 244.
3. Integration by Parts Calculus
If the integrand is a product of functions, then integration by parts is the best integral trick to use. Integration by parts reverses the product rule for derivatives. The formula of integration by parts is:
To use this formula, we can simply separate the integrand into a product of functions by choosing one function to represent u and one function to represent dv. It’s best to designate u as the term that’s easiest to differentiate, and dv as the term that’s easiest to integrate.
The next step is to differentiate u to find du and integrate dv to find v. Then, all that’s left to do is plug u, v, and du into the integration by parts formula and solve, simplifying where needed.
How do you know how to assign u and du? When deciding which function to appoint as u, one handy trick is to use the acronym LIATE:
Inverse trigonometric functions
Identify the different function types that are present in your integrand. The function types that are placed highest in the above list should be prioritized as u.
For example, let’s solve ∫ln(x)dx. Let u=ln(x) and dv=1dx. Differentiating u, we find that du=x1dx. Integrating dv, we find that v=x. Now, plugging u, v, and du into the integration by parts formula:
If the integrand is a composite function, then u-substitution is the best integral trick to use. By using u-substitution, we can easily reverse the chain rule for derivatives. To use this trick, we rewrite our integral in terms of u and du:
Make sure to replace all forms of x in your substitution. To do this, you might need to algebraically manipulate the problem a little bit; this might involve rewriting an expression in terms of u, or dividing or multiplying the integral by a constant. It can be helpful to solve for dx in terms of du to determine if this is the case.
For example, let’s integrate (7−x)4. Let u=7−x. Then du=−1dx, so dx=−1du. After substituting these values into our integrand, we can use the power rule. Remember to substitute the original values back into your final answer.
5. Trigonometric Identities
If the integrand involves trigonometric functions, one helpful trick is to use trigonometric identities to transform the integrand into a form that is more easily integrated.
Here are some of the most commonly used trigonometric identities:
Quotient and Reciprocal Identities
Double Angle Identities
6. Integrating Even and Odd Functions
For some definite integral problems, identifying if a function is even or odd can quickly simplify the problem.
If f(−x)=f(x), the function is even.
If f(−x)=−f(x), the function is odd.
After identifying that one of the above cases is true, examine the integral bounds. Determine if the lower and upper bounds are in the form −a and a. If so, then the following tricks can be used:
If f(x) is even, then ∫−aaf(x)dx=2∫0af(x)dx.
If f(x) is odd, then ∫−aaf(x)dx=0.
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