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Indefinite Integral

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
5. Practice Questions
6. FAQs on Indefinite Integral

What Is Indefinite Integral?

Indefinite Integrals are the reverse process of differentiation and are referred as the antiderivative of a function. 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.

d/dx.f(x) = f'(x)

\(\int f'(x).dx = f(x) + C \)

Indefinite Integral Formula

 

Important Formulas of Indefinite Integrals

Listed below are some of the important formulas of indefinite integrals.

  • \(\int x^n.dx = \dfrac{x^{n + 1}}{n + 1} + C \)

  • \(\int 1.dx = x + C \)

  • \(\int e^x.dx = e^x + C \)

  • \(\int\frac{1}{x}.dx = Log|x| + C \)

  • \(\int a^x.dx = \dfrac{a^x}{loga} + C \)

  • \(\int Cosx.dx = Sinx + C \)

  • \(\int Sinx.dx = -Cosx + C \)

  • \(\int Sec^2x.dx = Tanx + C \)

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 \(\int^b_af(x)dx\), 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).

Difference Between Indefinite Integral and Definite Integral

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.

Related Topics

The following topics help in a better understanding of definite integrals.

 

Examples on Indefinite Integral

  1. Example 1: Find the indefinite integral of \(x^{2 \over 3}\).

    Solution:

    \(\int x^{2 \over 3}.dx = \dfrac{ x^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} + C\)

    = \(\frac{x^{5 \over 3}}{5 \over 3} + C\)

    Therefore, the integral of \(x^{2 \over 3} \) is equal to \(\frac{x^{5 \over 3}}{5 \over 3} + C\).

  2. Example 2: Find the Indefinite Integral of \(Tan^2x\).

    Solution:

    We need to find the integral of \(Tan^2x\).

    \(\int Tan^2x.dx = \int (Sec^2x - 1).dx\)

    = \(\int Sec^2x.dx - \int 1.dx\)

    = Tanx - x + C

    Therefore, the integral of \(Tan^2x\) is Tanx - x + C.

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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 Important Formulas of Indefinite Integrals?

Some of the important formulas of indefinite integrals are as follows.

  • \(\int x^n.dx = \dfrac{x^{n + 1}}{n + 1} + C \)
  • \(\int 1.dx = x + C \)
  • \(\int e^x.dx = e^x + C \)
  • \(\int\frac{1}{x}.dx = Log|x| + C \)

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.

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