# Expert Maths Tutoring in the UK

Integration of Constants|Integration Rules | Integration| Math Dot Com
Integration of Constants|Integration Rules | Integration| Math Dot Com

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Constant of Integration

Constant of Integration is added to the answer of the integration. This is written to represent the constant term of the original function, which could not be obtained through this anti-derivative process. The constant of integration can have arbitrary values and is written as +C.

 1 What Is Constant of Integration? 2 Properties of Constant of Integration 3 Examples on Constant of Integration 4 Practice Questions 5 FAQs on Constant of Integration

## What Is Constant of Integration?

The constant of integration is the constant ‘C’ added to the result of the integration. The constant of integration is used to represent the term of the original expression, which cannot be obtained from the antiderivative of the function. The given function f(x), on finding the derivatives is f'(x), which on integrating gives g(x). Here the function g(x), on the addition of the constant term ‘C’, is equalized to the original function f(x).

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

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

And g(x) + C = f(x).

The derivative of any function with a constant value is equal to zero. And getting back the constant value through the process of anti-derivative is not possible, and hence it is represented by a constant of integration. The inherent ambiguity in the formulas of integration is represented as a constant. Further, the anti-derivative is a non-unique inverse of the derivative, which can be understood from the value of C.

The constant of integration represented the general solution, and on attributing certain values to the constant + C, we can obtain the particular solution.

## Properties of Constant of Integration

The following properties of constant of integration are helpful for a better understanding of constant of integration.

• The constant of integration is an arbitrary constant and it can have any value.
• The sum or difference of two constants of integration is written as a single constant of integration.
• For logarithmic or trigonometric functions, the constant of integration even if it involves a logarithmic or a trigonometric function is always written as only C.
• The constant of integration is only used for indefinite integrals and is not used for definite integrals.
• Multiplying and dividing the integral also multiplies or divides the constant of integration by the same number, which is presented as a constant C.
• Even with different methods of integration, and for different assumed values of constant of integration, we use the same alphabet C to represent the constant of integration.

Related Topics

The following topics help in a better understanding of the constant of integration.

## Examples on Constant of Integration

1. Example 1: Find the general solution of the integral $$e^x + x^2$$, by applying the constant of integration.

Solution:

The given expression for integration is $$e^x + x^2$$.

$$\int (e^x + x^2).dx = e^x +C_1 + \frac{x^3}{3} + C_2 = e^x + \frac{x^3}{3} + C$$

Here C is the constant of integration.

Therefore, the general solution of the integral by applying the constant of integration is $$e^x + \frac{x^3}{3} + C$$.

2. Example 2: Find the integration of the algebraic expression $$\dfrac{x – 1}{x^2}$$, using the constant of integration.

Solution:

The given expression for integration is $$\dfrac{x – 1}{x^2}$$.

$$\int (\dfrac{x – 1}{x^2}).dx=\int \dfrac{x}{x^2}.dx – \int \dfrac{1}{x^2}.dx$$

= $$\int \dfrac{1}{x}.dx – \int \dfrac{1}{x^2}.dx = Logx + LogC_1 – (-\dfrac{1}{x} – C_2) = Logx + 1/x + C$$

Therefore, the integral of the given expression by using the constant of integration is Logx + 1/x + C.

## FAQs on Constant of Integration

### How Do You Find the Constant of Integration?

The constant of integration can have arbitrary values, which are represented as ‘+C’ in the answer of the integration of the given function. There is no particular value for the constant of integration.

### What Is the Constant of Integration Used For?

The constant of integration is used to represent the constant term of the original function, which cannot be obtained through the process of integration.

### Is C the Constant of Integration?

The alphabet ‘C’ is used to represent the constant of integration.

### Why Do We Add +C in Integration?

The derivative of the constant term of the given function is equal to zero. The process of integration, or the anti-derivative process cannot realize the constant term of the function, and hence it is represented as +C.

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