Antiderivative Formula: Rules & Examples

Antiderivatives of a function are used to find the antiderivatives of different combinations of differentiated functions. These derivatives are related to definite integrals using the Fundamental Theorem of Calculus. 

• Antiderivatives are also known as inverse derivatives, primitive functions, primitive integrals or indefinite integrals. 
• The term "anti" derivatives comes from the fact that integration is the opposite process of differentiation. 
• One can retrieve the original function by differentiating the antiderivative of a function.
• It is used to calculate the antiderivative of the sum and difference of functions, the product and quotient of functions, the scalar multiple of a function and the constant function.
• Antiderivatives are used for the calculation of the volume of any 3-D curve.
• Mathematically it can be represented as

∫ f(x) dx = F(x) + C

• Where F'(x) = f(x) and
• C is the integration constant

Key Terms: Antiderivatives, Antiderivatives Formulas, Antiderivatives Rules, Antiderivative Power Rule, Antiderivative of Exponential Function, Properties of Antiderivatives, Definite Antiderivatives, Indefinite Antiderivatives, Fundamental Theorem of Calculus

---

What is Antiderivative?

[Click Here for Sample Questions]

The inverse of a function's derivative is known as its antiderivative. An integral of a function is another name for the antiderivative. 

• Antiderivative is used to determine the relationship between position, velocity and acceleration.
• It is used to determine the antidifferentiation of algebraic, hyperbolic, exponential, trigonometric, and logarithmic functions.
• If a function's derivative is d/dx[f(x)] = F(x) + C, then f(x) is the antiderivative of [F(x) + C] dx = F(x) + C. 

Read Also:

---

Antiderivatives Formulas

[Click Here for Sample Questions]

Several formulas are employed to determine the solve antiderivative functions are as follows:

• ∫ xⁿ dx = x^{(n + 1)}/(n + 1) + C
• ∫ e^x dx = e^x + C
• ∫ 1/x dx = log |x| + C

---

Types of Antiderivatives

[Click Here for Sample Questions]

There are two types of antiderivatives present:

• Indefinite Antiderivatives
• Definite Antiderivatives

Indefinite Antiderivatives

The indefinite antiderivative, also known as the indefinite integral, is the antiderivative of a function for which the antiderivative's (integration's) limit is unknown. 

• The result of these derivatives is accompanied by a constant value, often C, known as the integration constant. 
• Assuming we have a function F(x) with f(x) as its derivative,

∫ f(x) dx = F(x) + C, 

• C = Antiderivative constant

Definite Antiderivatives

The antiderivative of any function inside a closed interval is known as the definite integral or definite antiderivative.

• In this case, the answer to the integration is some contact value, and the integration constant is absent. 
• A definite antiderivative is used to compute the area under a curve.
• If a function F(x) is defined on the closed interval [a, b], then its definite antiderivative can be expressed as follows if its derivative is f(x).

∫^a_b f(x) = [F(x)]^a_b = F(a) – F(b)

• Where a and b are closed interval for function F(x)

---

Calculating Antiderivative

[Click Here for Sample Questions]

A formal method or formula is required to calculate the antiderivatives. Use the procedures listed below to determine the antiderivative of any function. 

• Examine the provided integral, then make an educated judgment as to the derivative of the function whose antiderivative has to be determined.
• To find the antiderivative of any function, use the integration formulas directly.
• First, determine the type of integrals, as simple integral problems can be solved using direct integration rules.
• You can solve some integral problems using the substitution method.
• We can solve rational algebraic functions by applying the partial fraction method.
• Additionally, the antiderivative of a definite function is determined using the characteristics of the definite integral.

The table below represents some standard functions and their integrals. 

---

Antiderivatives Rules

[Click Here for Sample Questions]

Antiderivative formulas follow three rules to carry out mathematical operations. The three rules are as follows:

• Constant rule
• Sum rule
• Difference rule

Constant Rule

In this process, the scalar value can be determined using integrals under the constant rule.

∫kf(x)dx = k ∫ f(x)dx, here “k” is any constant.

• Since k is a constant, it can be placed before or after the integral sign and won't alter the equation.

Sum Rule

The integral of the sum of two functions equals the total of the integrals of those two functions.

• According to this rule, the function of the sum rule is given as:

∫(f(x) + j(x))dx = ∫ f(x)dx + ∫j(x)dx

Difference Rule

According to this rule, the difference between the integrals of two functions is equal to the integral of their difference.

∫(f(x) - j(x))dx = ∫ f(x)dx - ∫j(x)dx

---

Antiderivative Power Rule

[Click Here for Sample Questions]

The antiderivative power rule is used to solve simple integral problems and integrate expressions with radical values. 

• It is used for integrating exponents and polynomials involving exponents of a variable.
• Antiderivative Power Rule can mathematically be represented as:

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

---

Antiderivative of Trigonometric Functions

[Click Here for Sample Questions]

Finding the antiderivative of the trigonometric functions is simple and aids in the solution of many integration issues. These are the trigonometric functions' antiderivatives.

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

Certain functions have an antiderivative that yields inverse trigonometric functions values, which are 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^-1x + C

---

Properties of Antiderivatives

[Click Here for Sample Questions]

An antiderivative function has a number of properties that help solve complex problems easily. Some of the properties of antiderivatives are as follows:

• ∫-f(x)dx = -∫f(x)dx
• ∫ f(x) dx = ∫g(x) dx if ∫[f(x) – g(x)]dx = 0
• ∫ [k₁f₁(x) + k₂f₂(x) + …+k_nf_n(x)]dx] = k₁∫ f₁(x)dx + k₂∫ f₂(x)dx + … + k_n∫ f_n(x)dx

---

Things to Remember

• An antiderivative is the outcome achieved when a function is subjected to the reverse process of differentiation, i.e., integration.
• They become definite integrals when limits are applied to them using the Fundamental Theorem of Calculus.
• The antiderivative rules can be used to find the antiderivatives of various combinations of logarithmic, exponential, inverse trigonometric, algebraic, trigonometric, and hyperbolic functions.
• There are various antidifferentiation rules that match most differentiation rules.
• For a constant function f(x) = k, the antiderivative rule is ∑k dx = kx + C.
• There are several antiderivatives of a given function that vary by a constant.
• The power rule, denoted as ∫ xn dx = xn + 1/(n + 1) + C, is the most significant antiderivative rule.

Read More: