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Indefinite integral is the integration of a function without any limits. Integration is the reverse process of differentiation and is referred as the antiderivative of the function. The indefinite integral is an important part of calculus and the application of limiting points to the integral transforms it to definite integrals. Integration is defined for a function f(x) and it helps in finding the area enclosed by the curve, with reference to one of the coordinate axes.
Indefinite integrals are further solved through different methods of integration by parts, integration by substitution, integration of partial fractions, and integration of inverse trigonometric functions. Let us learn more about indefinite integrals, important formulas, examples, and the difference between indefinite integrals and definite integrals.
|1.||What is Indefinite Integral?|
|2.||Important Formulas of Indefinite Integrals|
|3.||Difference Between Indefinite Integral and Definite Integral|
|4.||Examples on Indefinite Integral|
|6.||FAQs on Indefinite Integral|
What is Indefinite Integral?
Indefinite Integrals are the integrals that can be calculated by the reverse process of differentiation and are referred to as the antiderivatives of functions. For a function f(x), if the derivative is represented by f'(x), the integration of the resultant f'(x) gives back the initial function f(x). This process of integration can be defined as definite integrals. Let us understand this from the below expression.
- If d/dx f(x) = f'(x) then ∫ f'(x) dx = f(x) + C
Here, C is the constant of integration, and here is an example of why we need to add it after the value of every indefinite integral.
Example: Let f(x) = x2 and by power rule, f '(x) = 2x. Then the integral of f '(x) is, x2 + C, because by differentiating not only just x2 but also the functions such as x2 + 2, x2 - 1, etc gives 2x. The indefinite integral is techinically defined as shown below.
In the above definition:
- f(x) is called as the integrand
- dx means that the variable of integration is x
- F(x) is the value of the indefinite integral
i.e., the indefinite integral of a function f(x) is F(x) + C where, the derivative of F(x) is the original function f(x).
Calculate Indefinite Integral
The process of calculating indefinite integral depends on the given function. Here are the steps to calculate the indefinite integrals of different types of functions:
- Easy indefinite integrals can be solved by using direct integration formulas which are mentioned in the section below.
- Rational functions can be solved using the partial fractions method. i.e., we split the integrand using the partial fractions and then integrate each fraction separately.
- Some indefinite integrals can be solved by the substitution method.
- If the integrand is a product then it can be solved by using integration by parts.
- To evaluate a definite integral, evaluate the antiderivative first using one of the above methods and then apply the limits using the formula ∫ab f(x)dx = F(b) - F(a).
Example: Calculate the indefinite integral ∫ 3x2 sin x3 dx.
The given integral can be evaluated using the substitution method. Let us assume that x3 = t, then 3x2 dx = dt. Then the given integral becomes ∫ sin t dt. By using one of the rules of integration, its value is - cos t + C. Substituting t = x3 back, the value of the given indefinite integral is - cos x3 + C.
Important Formulas of Indefinite Integrals
Listed below are some of the important formulas of indefinite integrals. To learn more about these formulas and for more rules, click here.
∫ xn dx = xn + 1/ (n + 1) + C
∫ 1 dx = x + C
∫ ex dx = ex + C
∫1/x dx = ln |x| + C
∫ ax dx = ax / ln a + C
∫ cos x dx = sin x + C
∫ sin x dx = -cos x + C
∫ sec2x dx = tan x + C
Properties of Indefinite Integral
We may need to apply the properties below while evaluating an indefinite integral.
- Property of Sum: ∫ [f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx
- Property of Difference: ∫ [f(x) - g(x)]dx = ∫ f(x)dx - ∫ g(x)dx
- Property of Constant Multiple: ∫ k f(x)dx = k∫ f(x)dx
- ∫ f(x) dx = ∫ g(x) dx if ∫ [f(x) - g(x)]dx = 0
- ∫ [k1f1(x) + k2f2(x) + ...+knfn(x)]dx = k1∫ f1(x)dx + k2∫ f2(x)dx + ... + kn∫ fn(x)dx
Difference Between Indefinite Integral and Definite Integral
The indefinite integral is used to find the integral of the function, and the resultant expression represents the area enclosed by the function with reference to one of the axes. A definite integral has a defined value. The definite integral is represented as ∫ba f(x)dx, where a is the lower limit and b is the upper limit, for a function f(x), defined with reference to the x-axis. The definite integrals are the antiderivative of the function f(x) to obtain the function F(x), and the upper and lower limit is applied to find the value F(b) - F(a). This follows from the fundamental theorem of the caluculus.
Further, the numerous formulas and theorems used across indefinite integral can be used with definite integrals. The primary difference between indefinite integrals and definite integrals is that indefinite integrals do not have any limits, and there is an upper limit and lower limit in definite integrals.
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Indefinite Integral Examples
Example 1: Find the indefinite integral of x2/3.
∫ x2/3 dx = x2/3+1 / (2/3 + 1) + C
= x5/3 / (5/3) + C
= 3x5/3/ 5 + C
Answer: Therefore, the integral of x2/3 is equal to 3x5/3/ 5 + C.
Example 2: Find the indefinite integral of tan2x.
We need to find the integral of tan2x.
∫ tan2x dx = ∫ (sec2x - 1) dx
= ∫ sec2x dx - ∫ 1 dx
= tan x - x + C
Answer: Therefore, the integral of tan2x is tan x - x + C.
Example 3: Evaluate the indefinite integral ∫ sin2x+ cos2x dx.
By one of the trigonometric identities,
sin2x+ cos2x = 1.
Thus, the given indefinite integral becomes
∫ 1 dx = x + C
Answer: Therefore, ∫ sin2x+ cos2x dx = x + C.
Practice Questions on Indefinite Integral
FAQs on Indefinite Integral
What is the Difference Between Indefinite Integrals and Definite Integrals?
The definite integral and indefinite integrals differ by the application of limiting points. Indefinite integrals, we apply the lower limit and the upper limit to the points, and in indefinite integrals are computed for the entire range without any limits.
What Are the Properties of Indefinite Integrals?
The properties of indefinite integrals are similar to the differentiation properties. Indefinite integrals are the reverse of differentiation. The differentiation of f(x) gives f'(x), which on applying the indefinite integral gives back the function f(x).
What are the Methods Used to Evaluate Indefinite Integrals?
We use the following methods to calculate/evaluate the indefinite integrals:
- Using integration formulas
- Using u-substitution method
- Using uv method
- By Splitting into partial fractions
- By applying trig formulas
What Are the Important Rules of Indefinite Integrals?
Some of the important rules of indefinite integrals are as follows.
- ∫ xn dx = xn + 1/ (n + 1) + C
- ∫ 1 dx = x + C
- ∫ ex dx = ex + C
- ∫ 1/x dx = ln |x| + C
What Are the Applications of Indefinite Integrals?
The topic of indefinite integral has numerous applications in calculus. The concept of indefinite integral can be used to find the area enclosed by the given equation of the curve. Further on applying limits the indefinite integral are transformed into a definite integral, which help in the calculation of the area enclosed by this curve.
What is the Indefinite Integral of e^x?
The integral of e to the power of x is itself. But we add an integration constant to every indefinite integral. i.e., ∫ ex dx = ex + C