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.
|1.||What is Antiderivative?|
|5.||Antiderivative of Trig Functions|
|6.||Antiderivative of Exponential Function|
|7.||Properties of Antiderivatives|
|8.||FAQs on Antiderivatives|
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
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
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:
∫ab f(x) = [F(x)]ab = F(b) - F(a)
This follows from the fundamental theorem of calculus.
There are several different antiderivative formulas that help to find the antiderivative of a given function using the process of integration. These help to increase the speed and accuracy of performing calculations. Some antiderivative formulas are given below:
- ∫ xn dx = xn + 1/(n + 1) + C
- ∫ ex dx = ex + C
- ∫ 1/x dx = log |x| + C
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 ∫ab f(x)dx = F(b) - F(a) to get the final answer.
There are certain important rules that need to be followed while integrating a function to obtain its antiderivatives. These rules are listed as follows:
- Sum Rule: The antiderivative of a sum is equal to the sum of the antiderivatives. If f(x) and g(x) are two functions then ∫ [f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx
- Difference Rule: This rule states that the antiderivative of a difference is equal to the difference of the antiderivatives. This can be expressed as ∫ [f(x) - g(x)]dx = ∫ f(x)dx - ∫ g(x)dx
- Constant Rule: A scalar can be taken out of an integral under the constant rule. If k is a scalar or a constant then ∫ k f(x)dx = k∫ f(x)dx
The most important rule is the power rule that will be studied in the upcoming section.
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 = xn + 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 = x3 + 1/(3 + 1) = x4 / 4 + C is the antiderviative of x3.
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, ex, 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 ex is, ∫ ex dx = ex + C
- ∫ ecx dx = (1/c) ecx + C
Suppose a constant number is raised to the exponent x then the antiderivative of such a function is as follows:
Antiderivative of ax is, ∫ ax dx = (1 / ln a) ax + 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) + ...+knfn(x)]dx = k1∫ f1(x)dx + k2∫ f2(x)dx + ... + kn∫ fn(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 = xn + 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 = ∫ -et dt = -et + 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
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)
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 = xn + 1/(n + 1) + C
∫ 5x4 dx = 5x4 + 1/ (4+1) + C
= x5 + C
Answer: ∫ 5x4 dx = = x5 + 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. To learn these methods in detail, click here.
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 = xn + 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.