Antiderivative Rules

The antiderivative rules in calculus are basic rules that are used to find the antiderivatives of different combinations of functions. As the name suggests, antidifferentiation is the reverse process of differentiation. These antiderivative rules help us to find the antiderivative of sum or difference of functions, product and quotient of functions, scalar multiple of a function and constant function, and composition of functions. The antiderivative rules help us to make the process of finding the antiderivatives easier and simpler.

Further in this article, we will proceed with exploring the antiderivative rules in detail. We will also go through the antiderivative rules of some important and specific functions along with some solved examples for a better understanding of the concept.

Antiderivative rules are some of the important rules to find the antiderivatives of different forms of combinations of a function. We can use these antiderivative rules to find the antiderivatives of product, quotient, sum, difference, scalar multiple, and composition of functions. These rules can be used for the antidifferentiation of algebraic functions, exponential function, trigonometric functions, hyperbolic functions, logarithmic function, and constant function. Let us go through the important antiderivative rules in the sections below.

The list of most commonly used antiderivative rules for the product, quotient, sum, difference, and the composition of functions is as follows:

• Antiderivative Power Rule
• Antiderivative Chain Rule
• Antiderivative Product Rule
• Antiderivative Quotient Rule
• Antiderivative Rule for Scalar Multiple of Function
• Antiderivative Rule for Sum and Difference of Functions

In this section, we will explore the formulas for the different antiderivative rules discussed above in detail. We will discuss the rules for the antidifferentiation of algebraic functions with power, and various combinations of functions. The antiderivative rules are common for types of functions such as trigonometric, exponential, logarithmic, and algebraic functions.

Antiderivative Power Rule

Now, the antiderivative rule of power of x is given by ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, where n ≠ -1. This rule is commonly known as the antiderivative power rule. Let us consider some of the examples of this antiderivative rule to understand this rule better.

• ∫x² dx = x²⁺¹/(2+1) + C = x³/3 + C
• ∫x^-4 dx = x^-4+1/(-4+1) + C = x^-3/(-3) + C = -x^-3/3 + C

Using the antiderivative power rule, we can conclude that for n = 0, we have ∫x⁰ dx = ∫1 dx = ∫dx = x⁰⁺¹/(0+1) + C = x + C. Please do not confuse this power antiderivative rule ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, where n ≠ -1 with the power rule of derivatives which is d(xⁿ)/dx = nx^n-1.

Antiderivative Chain Rule

We know that antidifferentiation is the reverse process of differentiation, therefore the rules of derivatives lead to some antiderivative rules. The chain rule of derivatives gives us the antiderivative chain rule which is also known as the u-substitution method of antidifferentiation. The antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x)) dx. Let us see an example and solve an integral using this antiderivative rule.

Example: Solve ∫2x cos (x²) dx

Solution: Assume x² = u ⇒ 2x dx = du. Substitute this into the integral, we have

∫2x cos (x²) dx = ∫cos u du

= sin u + C

= sin (x²) + C

Antiderivative Product Rule

The antiderivative product rule is also commonly called the integration by parts method of integration. It is one of the important antiderivative rules and is used when the antidifferentiation of the product of functions is to be determined. The formula for the antiderivative product rule is ∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C. The choice of the first function is done on the basis of the sequence given below. This method is also commonly known as the ILATE or LIATE method of integration which is abbreviated of:

• I - Inverse Trigonometric Function
• L - Logarithmic Function
• A - Algebraic Function
• T - Trigonometric Function
• E - Exponential Function

For example, we need to find the antiderivative of x ln x. Then, according to the sequence above, the first function is ln x and the second function is x. Therefore, we have

∫x ln x dx = ln x ∫x dx - ∫[(ln x)' ∫x dx] dx

= (x²/2) ln x - ∫(1/x)(x²/2) dx

= (x²/2) ln x - ∫(x/2) dx

= (x²/2) ln x - x²/4 + C

Antiderivative Quotient Rule

The antiderivative quotient rule is used when the function is given in the form of numerator and denominator. If the function includes algebraic functions, then we can use the integration by partial fractions method of antidifferentiation. Another way to determine the antiderivative of the quotient of functions is, consider a function of the form f(x)/g(x). Now, differentiating this we have,

d(f(x)/g(x))/dx = [f'(x)g(x) - g'(x)f(x)]/[g(x)]²

Now, integrating both sides of the above equation, we have

f(x)/g(x) = ∫{[f'(x)g(x) - g'(x)f(x)]/[g(x)]²} dx

= ∫[f'(x)/g(x)] dx - ∫[f(x)g'(x)/[g(x)]²] dx

⇒ ∫[f'(x)/g(x)] dx = f(x)/g(x) + ∫[f(x)g'(x)/[g(x)]²] dx

If f(x) = u and g(x) = v, then we have the antiderivatiev quotient rule as:

∫du/v = u/v + ∫[u/v²] dv

Antiderivative Rule for Scalar Multiple of Function

To find the antiderivative of scalar multiple of a function f(x), we can find it using the formula given by, ∫kf(x) dx = k ∫f(x) dx. This implies, the antidifferentiation of kf(x) is equal to k times the antidifferentiation of f(x), where k is a scalar. An example using this antiderivative rule is:

∫4x dx = 4 ∫xdx

= 4 × x²/2 + C

= 2x² + C

Sum and Difference Antiderivative Rule

Now, this rule is one of the easiest antiderivative rules. When the antidifferentiation of the sum and difference of functions is to be determined, then we can do it by using the following formulas:

• ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
• ∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx

Some of the examples of the antiderivative rule for sum and difference of functions are as follows:

• ∫[4 + x²] dx = ∫4 dx + ∫x² dx = 4x + x³/3 + C
• ∫(sin x - log x) dx = ∫sin x dx - ∫ log x dx = -cos x - x log x + x + C

To use the antiderivative rules, we must know the antiderivatives of some specific functions such as the exponential function, trigonometric functions, logarithmic functions, hyperbolic functions, and inverse trigonometric functions. Let us go through the antiderivative rules for these functions:

Antiderivative Rules for Trigonometric Functions

We have six main trigonometric functions, namely sine, cosine, tangent, cotangent, secant, and cosecant. Now, we will explore their antiderivative rules of these trigonometric functions as follows:

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫tan x dx = ln |sec x| + C
• ∫cot x dx = ln |sin x| + C
• ∫sec x dx = ln |sec x + tan x| + C
• ∫csc x dx = ln |cosec x - cot x| + C

Antiderivative Rules for Inverse Trigonometric Functions

We have six main inverse trigonometric functions, namely inverse sine, inverse cosine, inverse tangent, inverse cotangent, inverse secant, and inverse cosecant. Now, we will explore their antiderivative rules of these trigonometric functions as follows:

• ∫sin^-1x dx = x sin^-1x + √(1 - x²) + C
• ∫cos^-1x dx = x cos^-1x - √(1 - x²) + C
• ∫tan^-1x dx = x tan^-1x - (1/2) ln(1 + x²) + C
• ∫cot^-1x dx = x cot^-1x + (1/2) ln(1 + x²) + C
• ∫sec^-1x dx = x sec^-1x - ln |x + √(x² - 1)| + C
• ∫csc^-1x dx = x csc^-1x + ln |x + √(x² - 1)| + C

Antiderivative Rules for Exponential Functions

The exponential function is of the form f(x) = a^x, where a is the base (real number) and x is the variable. When a is equal to the Euler's number e, then we have f(x) = e^x, where e is a constant whose value is approximately 2.718. Now, the antiderivative rules for these two forms of the exponential functions are:

• ∫a^x dx = a^x/ln a + C
• ∫e^x dx = e^x + C [Because ln e = 1]

Antiderivative Rules for Logarithmic Functions

The logarithmic function is generally of the form f(x) = log_ax, where a is the base and x is the variable. If the base a is equal to the Euler's number e, then it is called the natural logarithmic function and is written as f(x) = ln x. The antiderivative rules for the logarithmic function are:

• ∫log_ax dx = x log_ax - x/ln a + C
• ∫ln x dx = x ln x - x + C

Antiderivative Rules for Hyperbolic Functions

Now, the hyperbolic functions are analogous to the trigonometric functions but they are derived using a hyperbola instead of a unit circle as in the case of trigonometric functions. The six main hyperbolic functions are sinh x, cosh x, tanh x, coth x, sech x, and csch x. The antiderivative rules of hyperbolic functions are:

• ∫sinh x dx = cosh x + C
• ∫cosh x dx = sinh x + C
• ∫tanh x dx = ln (cosh x) + C
• ∫coth x dx = ln (sinh x) + C
• ∫sech x dx = arctan(sinh x) + C
• ∫csch x dx = ln(tanh (x/2)) + C

Important Notes on Antiderivative Rules

• The antiderivatives rules are used to find the antiderivatives of different combinations of algebraic, trigonometric, logarithmic, exponential, inverse trigonometric, and hyperbolic functions.
• Most of the rules of differentiation have corresponding antiderivative rules for antidifferentiation.
• The antiderivative rule for a constant function f(x) = k is ∫k dx = kx + C.

☛ Related Articles:

• Antiderivative Formula
• Antiderivative of Ln x
• Integral Calculus

What are Antiderivative Rules in Calculus?

Antiderivative rules are some of the important rules in calculus that are used to find the antiderivatives of different forms of combinations of a function. These antiderivative rules help us to find the antiderivative of sum or difference of functions, product and quotient of functions, scalar multiple of a function and constant function, and composition of functions.

How Do You Use Antiderivative Chain Rule?

The antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x)) dx. It is commonly known as the u-substitution method of antidifferentiation. We generally substitute the function u(x) by assuming it be another variable.

What are the Commonly Used Antiderivative Rules?

Most commonly used antiderivative rules for the product, quotient, sum, difference, and the composition of functions are as follows:

• Antiderivative Power Rule
• Antiderivative Chain Rule
• Antiderivative Product Rule
• Antiderivative Quotient Rule
• Antiderivative Rule for Scalar Multiple of Function
• Antiderivative Rule for Sum and Difference of Functions

What are the Antiderivative Rules for Trig Functions?

The antiderivative rules of the six trigonometric functions are as follows:

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫tan x dx = ln |sec x| + C
• ∫cot x dx = ln |sin x| + C
• ∫sec x dx = ln |sec x + tan x| + C
• ∫csc x dx = ln |cosec x - cot x| + C

How To Use Antiderivative Rules for Exponential Functions?

The antiderivative rules for the two forms of the exponential functions are:

• ∫a^x dx = a^x/ln a + C
• ∫e^x dx = e^x + C

What is The Antiderivative Product Rule?

The formula for the antiderivative product rule is ∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C where we need to find the antiderivative of the product of two or more functions.