Antiderivative Formula

Anything that is the opposite of a function and has been differentiated in trigonometric terms is known as an anti-derivative. Both the antiderivative and the differentiated function are continuous on a specified interval. In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. Some of the formulas are mentioned below.

Basic Antiderivatives

These can be written using integral as given below:

Antiderivative Rules

List of Antiderivative Rules

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

Antiderivative Power Rule

The antiderivative power rule is also the general formula that is used to solve simple integrals. It shows how to integrate a function of the form xⁿ, where n ≠ -1. This rule can also be used to integrate expressions with radicals in them. The power rule for antiderivatives is given as follows:

∫ xⁿ dx = x^{n + 1}/(n + 1) + C, where C is the integration constant.

Suppose there is a function x³. Then as the power of the function is 3, which is not equal to -1, the power rule can be used to integrate it. ∫ x³ dx = x^{3 + 1}/(3 + 1) = x⁴ / 4 + C is the antiderviative of x³.

Basic Antiderivative Rules

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 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

Antiderivative Rules for Specific Functions

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.

Antiderivative

Antiderivative of functions is also known as integral. When the antiderivative of a function is differentiated, the original function is obtained. Integration is the opposite process of differentiation and hence the name “anti” derivatives.

Antiderivatives are usually known as indefinite integrals. However, using the Fundamental Theorem of Calculus antiderivatives can also be related to definite integrals. In this article, we will learn about antiderivatives, their formulas, rules, and various applications.

What is Antiderivative?

An antiderivative, F, of a function, f, can be defined as a function that can be differentiated to obtain the original function, f. i.e., an antiderivative is mathematically defined as follows: ∫ f(x) dx = F(x) + C, where

• the derivative of F(x) is f(x). i.e., F'(x) = f(x) and
• C is the integration constant

A given function can have many antiderivatives and thus, they are not unique. The antiderivatives of a function x could be x²/2 + 2, x²/2 – 32, x²/2 + 19.2, and so on (try to differentiate each of these and find the result to be x). Thus, it can be said that antiderivatives of a function will differ by a constant. Antiderivatives can be further classified into two types :

• indefinite antiderivatives
• definite antiderivatives

Indefinite Antiderivative

When the general antiderivative of a function is determined it is known as an indefinite antiderivative (or) indefinite integral. Such an antiderivative does not have any limits/bounds. Integration, which is the reverse process of differentiation, is used to calculate the indefinite antiderivative of a function. Suppose there is a function f(x) and its antiderivative if F(x). It is written as follows:

∫ f(x) dx = F(x) + C

where C is a real number and is the constant of integration. ‘∫’ is the integral sign.

Definite Antiderivative

If the antiderivative of a function is evaluated between two endpoints then it is known as a definite antiderivative (or) definite integral. The definite integral of a function is used to compute the area under a curve. Such an antiderivative will have a definite value. Suppose an antiderivative of a function, f(x), has to be evaluated between two points (or limits) a and b then it is written as follows:

∫_a^b f(x) = [F(x)]_a^b = F(b) – F(a)

This follows from the fundamental theorem of calculus.

Calculating Antiderivative

The process of calculating antiderivative depends on the complexity of the function. The steps to calculate the antiderivatives of different types of functions are listed below:

• Check the type of integral. Easy integrals can be solved by using direct integration rules.
• Some integrals can be solved by the substitution method.
• Rational algebraic functions can be solved using the integration by partial fractions method.
• Functions expressed as a product can be solved by using integration by parts.
• For a definite integral, evaluate the antiderivative first using one of the above examples and then apply the limits using the formula ∫_a^b f(x)dx = F(b) – F(a) to get the final answer.

Antiderivative of Trig Functions

There are six basic trigonometric functions. These are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). cosec, sec and cot are reciprocal functions of sin, cos and tan respectively. The antiderivatives of trigonometric functions are given below:

• Antiderivative of sin x is, ∫ sin x dx = -cos x + C
• Antiderivative of cos x is, ∫ cos x dx = sin x + C
• Antiderivative of tan x is, ∫ tan x dx = -ln |cos x| + C = ln |sec x| + C
• Antiderivative of cot x is, ∫ cot x dx = ln |sin x| + C = -ln |cosec x| + C
• Antiderivative of sec x is, ∫ sec x dx = ln |sec x + tan x| + C
• Antiderivative of cosec x is, ∫ cosec x dx = – ln |cosec x + cot x| + C
• ∫ cos (ax + b)x dx = (1/a) sin (ax + b) + C
• ∫ sin (ax + b)x dx = -(1/a) cos (ax + b) + C

There are certain functions that give inverse trigonometric functions as the antiderivatives on integration. These are given as follows:

• ∫1/√(1 – x²).dx = sin^-1x + C
• ∫ 1/(1 – x²).dx = -cos^-1x + C
• ∫1/(1 + x²).dx = tan^-1x + C
• ∫ 1/(1 + x² ).dx = -cot^-1x + C
• ∫ 1/x√(x² – 1).dx = sec^-1x + C
• ∫ 1/x√(x² – 1).dx = -cosec^-1 x + C

Apart from these, we have reduction formulas that talk about the antiderivatives of sinⁿx, cosⁿx, and tanⁿx.

Antiderivative of Exponential Function

Exponential functions are widely used to model situations such as financial growth, population growth., etc. This is because, e, is usually associated with accelerating or compounding growth. An exponential function, e^x, is its own antiderivative and derivative. The power rule cannot be used to integrate an exponential function. The antiderivative of an exponential function is given as follows:

• Antiderivative of e^x is, ∫ e^x dx = e^x + C
• ∫ e^cx dx = (1/c) e^cx + C

Suppose a constant number is raised to the exponent x then the antiderivative of such a function is as follows:

Antiderivative of a^x is, ∫ a^x dx = (1 / ln a) a^x + C

Another important formula that falls under the category of exponential functions is the antiderivative of a logarithmic function. A logarithmic function can be integrated using the following formulas:

• Antiderivative of log x is, ∫ log x dx = xlog x – x + C
• Antiderivative of ln x is, ∫ ln x dx = x ln x – x + C.

Properties of Antiderivatives

The properties of antiderivatives help to simplify an otherwise complicated expression so as to make calculations easier. Some important properties of antiderivatives are as follows:

• ∫ f(x) dx = ∫ g(x) dx if ∫ [f(x) – g(x)]dx = 0. This is a consequence of the difference rule.
• ∫ [k₁f₁(x) + k₂f₂(x) + …+k_nf_n(x)]dx = k₁∫ f₁(x)dx + k₂∫ f₂(x)dx + … + k_n∫ f_n(x)dx. This property is a consequence of the sum rule and the constant rule.

Important Notes on Antiderivatives:

• On applying the reverse process of differentiation, i.e., integration, to a function the result so obtained is known as an antiderivative.
• Add a constant C after finding any antiderivative.
• A given function can have multiple antiderivatives that differ by a constant.
• The power rule is the most important antiderivative rule given by ∫ xⁿ dx = x^{n + 1}/(n + 1) + C
• An antiderivative is an indefinite integral. When limits are applied to antiderivatives, using the Fundamental Theorem of Calculus, they become definite integrals.

Examples on Antiderivatives

Example 1: Calculate the antiderivative of e^-x.
Solution: Using substitution t = -x.
dt = -dx or dx = -dt
∫ e^-x dx = ∫ -e^t dt = -e^t + C = -e^-x + C
Answer: ∫ e^-x dx = -e^-x + C

Example 2: Find the antiderivative of f(x) = 2x cos (x² + 1). Also, verify the antiderivative by differentiation.
Solution: Using substitution t = x² + 1.
dt = 2x dx
∫ 2x cos (x² +1)dx = ∫ cos t dt = sin t + C
= sin (x² + 1) + C
Verification:
Let us find the derivative of the above result.
d/dx (sin (x² + 1) + C) = cos (x² + 1) d/dx (x² + 1) (by chain rule)
= 2x cos (x² + 1)
= f(x)
Hence, the antiderivative is verified.
Answer: ∫ 2x cos (x² +1) = sin (x² + 1) + C

Example 3: Calculate the antiderivative of 5x⁴.
Solution: Using the antiderivative power rule,
∫ xⁿ dx = x^{n + 1}/(n + 1) + C
∫ 5x⁴ dx = 5x^{4 + 1}/ (4+1) + C
= x⁵ + C
Answer: ∫ 5x⁴ dx = = x⁵ + C

Example 1: Evaluate the antiderivative of f(x) = 10(x² – x – 7) using the antiderivative rules.

Solution: To find the antiderivative of f(x) = 10(x² – x – 7), we will use the following antiderivative rules:

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) → Antiderivative Power Rule∫kf(x) dx = k ∫f(x) dx → Antiderivative Rule of Scalar Multiplication∫[f(x) – g(x)] dx = ∫f(x) dx – ∫g(x) dx → Antiderivative Rule of Difference

∫10(x² – x – 7) dx = ∫10x² dx – ∫10x dx – ∫70 dx= 10 ∫x² dx – 10 ∫x dx – 70 ∫ dx= 10 x³/3 – 10 x²/2 – 70x + C= (10/3)x³ – 5x² – 70x + C

Answer: The antiderivative of f(x) = 10(x² – x – 7) is (10/3)x³ – 5x² – 70x + C.

Example 2: Determine the antiderivative of f(x) = xe^x using the antiderivative rules.

Solution: To find the antiderivative of f(x) = xe^x, we will use the antiderivative product rule as f(x) is a product of two functions x and e^x. Therefore, we have∫xe^x dx = x ∫e^x dx – ∫[dx/dx × ∫e^x dx] dx= xe^x – ∫1e^x dx= xe^x – e^x + C= e^x (x – 1) + C

Answer: Therefore, the antiderivative of f(x) = xe^x is e^x (x – 1) + C.

FAQs on Antiderivatives

What are Antiderivatives?

Antiderivatives are the functions that are obtained after integrating a given function. Antiderivatives are a part of integral calculus. If an antiderivative is differentiated, the original function is obtained.

What is the Purpose of Antiderivatives?

The process that reverses the outcome of differentiation is known as the antiderivative. A function can be integrated to get the antiderivative and a constant of integration.

How to Find Antiderivatives?

To find antiderivatives, integrate the given function using formulas, substitution method, integration by parts, or integration by partial fractions. The final result will have a constant of integration if no limits are specified in the original function.

Are Antiderivatives the Same as Integrals?

Antiderivatives are the same as indefinite integrals. However, if certain limits are specified in the given function then the antiderivative works as a definite integral.

What are the Methods to Calculate Antiderivatives?

Some antiderivatives can be calculated just by applying antiderivative rules. But for calculating some antiderivatives, we need methods like substitution method, integration by parts, integration by partial fractions, etc.

What is the Power Rule for Antiderivatives?

The power rule for antiderivatives is applied to functions of the form xⁿ where n is not equal to -1. It is given as ∫ xⁿ dx = x^{n + 1}/(n + 1) + C.

What is the Antiderivative of 1 / x?

The antiderivative of 1 / x is ln|x| + C. This is because the derivative of ln x is 1/x.

What are the Applications of Antiderivatives?

Antiderivatives are widely used to explain the relationship between speed, position, and velocity. For example, integration of acceleration results in the velocity of a moving object along with a constant.

What is an antiderivative in math?

The antiderivative is the function that you can take the derivative of to get the original function. That is, the function F(x) is the antiderivative of f(x) if F'(x) = f(x).

What is the antiderivative used for?

An antiderivative can be used to “undo” a derivative, although some information is lost. It can also be used to find the area under a curve.

What is the difference between an integral and an antiderivative?

They are very similar. An antiderivative is the same thing as an indefinite integral. A definite integral, on the other hand, involves limits of integration.

What is the antiderivative of 2x?

The antiderivative of 2x is x^2 + C. You can find this by applying the product rule and the power rule.

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.

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