How To Find The 12th Derivative

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 procedure of differentiation and hence the name "anti" derivatives.

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

i.	What is Antiderivative?
two.	Antiderivatives Formulas
iii.	Calculating Antiderivative
four.	Antiderivatives Rule
5.	Antiderivative of Trig Functions
six.	Antiderivative of Exponential Function
7.	Properties of Antiderivatives
8.	FAQs on Antiderivatives

What is Antiderivative?

An antiderivative, F, of a function, f, tin can be defined equally a function that can be differentiated to obtain the original function, f. i.eastward., an antiderivative is mathematically defined as follows: ∫ f(x) dx = F(10) + C, where

• the derivative of F(10) is f(x). i.east., 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 office x could be x²/ii + 2, x²/2 - 32, x²/ii + 19.two, and then on (attempt to differentiate each of these and find the upshot to be x). Thus, it can exist said that antiderivatives of a function will differ by a constant. Antiderivatives can exist farther classified into 2 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 take whatsoever limits/premises. Integration, which is the reverse process of differentiation, is used to summate the indefinite antiderivative of a function. Suppose there is a function f(x) and its antiderivative if F(x). It is written every bit follows:

∫ f(x) dx = F(ten) + C

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

Definite Antiderivative

If the antiderivative of a office is evaluated between 2 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 betwixt two points (or limits) a and b then it is written as follows:

∫_a ^b f(10) = [F(ten)]_a ^b = F(b) - F(a)

This follows from the primal theorem of calculus.

Antiderivatives Formulas

There are several dissimilar antiderivative formulas that assistance to discover 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 beneath:

• ∫ 10ⁿ dx = x^{northward + i}/(due north + i) + C
• ∫ e^x dx = e¹⁰ + C
• ∫ ane/x dx = log |x| + C

Calculating Antiderivative

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

• Bank check the blazon of integral. Easy integrals can be solved by using directly integration rules.
• Some integrals can be solved by the exchange method.
• Rational algebraic functions can be solved using the integration by partial fractions method.
• Functions expressed equally a product can be solved past using integration by parts.
• For a definite integral, evaluate the antiderivative first using one of the to a higher place examples and so apply the limits using the formula ∫_a ^b f(ten)dx = F(b) - F(a) to go the final respond.

Antiderivatives Rules

There are certain important rules that demand to be followed while integrating a function to obtain its antiderivatives. These rules are listed equally follows:

• Sum Rule: The antiderivative of a sum is equal to the sum of the antiderivatives. If f(x) and yard(x) are two functions then ∫ [f(x) + g(10)]dx = ∫ f(ten)dx + ∫ g(x)dx
• Deviation Rule: This rule states that the antiderivative of a difference is equal to the difference of the antiderivatives. This can be expressed as ∫ [f(ten) - g(x)]dx = ∫ f(10)dx - ∫ g(x)dx
• Constant Rule: A scalar tin can be taken out of an integral nether the abiding rule. If m is a scalar or a constant and then ∫ k f(x)dx = thousand∫ f(10)dx

The most important rule is the power dominion 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. Information technology shows how to integrate a function of the course x^{due north}, where n ≠ -1. This rule can also be used to integrate expressions with radicals in them. The ability dominion for antiderivatives is given as follows:

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

Suppose in that location is a function x^three. Then as the power of the function is 3, which is not equal to -ane, the power rule tin be used to integrate information technology. ∫ x³ dx = ten^{3 + 1}/(3 + 1) = x⁴ / 4 + C is the antiderviative of x³.

Antiderivative of Trig Functions

There are 6 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 ten 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 ten| + C = -ln |cosec x| + C
• Antiderivative of sec x is, ∫ sec ten dx = ln |sec x + tan x| + C
• Antiderivative of cosec x is, ∫ cosec x dx = - ln |cosec x + cot ten| + C
• ∫ cos (ax + b)ten dx = (1/a) sin (ax + b) + C
• ∫ sin (ax + b)x dx = -(1/a) cos (ax + b) + C

There are sure functions that give changed trigonometric functions as the antiderivatives on integration. These are given equally follows:

• ∫1/√(i - x²).dx = sin^-1x + C
• ∫ 1/(1 - x²).dx = -cos^-1x + C
• ∫1/(i + ten²).dx = tan^-1x + C
• ∫ 1/(i + x² ).dx = -cot^-anex + C
• ∫ one/10√(ten² - 1).dx = sec^-110 + C
• ∫ 1/x√(xⁱⁱ - i).dx = -cosec^-1 ten + C

Apart from these, nosotros have reduction formulas that talk about the antiderivatives of sin^{due north}ten, cosⁿx, and tanⁿten.

Antiderivative of Exponential Office

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^ten, is its own antiderivative and derivative. The power rule cannot be used to integrate an exponential office. The antiderivative of an exponential function is given as follows:

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

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

Antiderivative of a^x is, ∫ a¹⁰ dx = (1 / ln a) a^x + C

Some other important formula that falls nether the category of exponential functions is the antiderivative of a logarithmic office. A logarithmic function can be integrated using the following formulas:

• Antiderivative of log x is, ∫ log 10 dx = xlog ten - 10 + C
• Antiderivative of ln x is, ∫ ln x dx = x ln ten - 10 + C.

Backdrop of Antiderivatives

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

• ∫ f(10) dx = ∫ thousand(10) dx if ∫ [f(10) - g(10)]dx = 0. This is a consequence of the deviation rule.
• ∫ [k₁f₁(ten) + k_iif_ii(x) + ...+k_northf_n(x)]dx = k₁∫ f_one(x)dx + 1000₂∫ f₂(x)dx + ... + k_n∫ f_n(x)dx. This property is a consequence of the sum rule and the constant rule.

☛Related Articles:

• Integration Formulas
• Differential Equations
• Trigonometry

Of import 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 together a constant C after finding whatsoever antiderivative.
• A given function can take multiple antiderivatives that differ past a constant.
• The ability rule is the nearly of import antiderivative rule given past ∫ xⁿ dx = x^{northward + ane}/(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

1. Case ane: Calculate the antiderivative of eastward^-x.
   Solution: Using substitution t = -x.
   dt = -dx or dx = -dt
   ∫ eastward^-ten dx = ∫ -e^t dt = -e^t + C = -eastward^-10 + C
   Answer: ∫ e^-x dx = -eastward^-10 + C

2. Example ii: Discover the antiderivative of f(10) = 2x cos (10² + 1). Also, verify the antiderivative past differentiation.
   Solution: Using substitution t = x^two + 1.
   dt = 2x dx
   ∫ 2x cos (x² +1)dx = ∫ cos t dt = sin t + C
   = sin (x² + one) + C
   Verification:
   Let us find the derivative of the to a higher place result.
   d/dx (sin (x² + one) + C) = cos (x^two + 1) d/dx (x² + 1) (past concatenation rule)
   = 2x cos (ten² + 1)
   = f(x)
   Hence, the antiderivative is verified.
   Answer: ∫ 2x cos (xⁱⁱ +1) = sin (x² + one) + C

3. Instance 3: Calculate the antiderivative of 5x⁴.
   Solution: Using the antiderivative power rule,
   ∫ 10ⁿ dx = ten^{due north + 1}/(n + 1) + C
   ∫ 5x^four dx = 5x^{4 + i}/ (iv+one) + C
   = x⁵ + C
   Answer: ∫ 5x⁴ dx = = ten⁵ + C

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Practice Questions on Antiderivatives

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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 role is obtained.

What is the Purpose of Antiderivatives?

The procedure that reverses the outcome of differentiation is known as the antiderivative. A part tin exist integrated to get the antiderivative and a constant of integration.

How to Notice Antiderivatives?

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

Are Antiderivatives the Same as Integrals?

Antiderivatives are the aforementioned as indefinite integrals. Even so, if certain limits are specified in the given function and so the antiderivative works every bit a definite integral.

What are the Methods to Calculate Antiderivatives?

Some antiderivatives can be calculated just by applying antiderivative rules. Only for calculating some antiderivatives, we need methods like commutation 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 dominion for antiderivatives is applied to functions of the grade 10ⁿ where n is not equal to -1. Information technology is given as ∫ xⁿ dx = ten^{n + 1}/(n + i) + C.

What is the Antiderivative of 1 / x?

The antiderivative of 1 / x is ln|ten| + C. This is because the derivative of ln 10 is 1/ten.

What are the Applications of Antiderivatives?

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

Source: https://www.cuemath.com/calculus/antiderivatives/

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