Question

What is Constant of Integration?

Answer 1

Constant of Integration:

• The constant of integration is the constant '$C$' added to the result of the integration.
• Because the derivative of any function with a constant value is equal to zero and recovering the constant value through the process of anti-derivative is therefore not possible, it is used to represent the term of the original expression, which cannot be obtained from the anti-derivative of the function. As a result, it is represented by an integration constant.
• The ambiguity present in the integration formulas is expressed as a constant.
• Consider the given function $f\left(x\right)$, on finding the derivative is $f\text{'}\left(x\right)$, which on integrating gives $g\left(x\right)$. Here the function $g\left(x\right)$, on the addition of the constant term '$C$', is equalized to the original function $f\left(x\right)$.

i.e. $\frac{d}{dx}\left[f\left(x\right)\right]=f\text{'}\left(x\right)$

$\int f\text{'}\left(x\right)dx=g\left(x\right)+C$ and $g\left(x\right)+C=f\left(x\right)$

Properties of Constant of Integration

The following are the properties of constant of integration:

1. The integration constant is a freely chosen constant that can be any value.
2. A single integration constant is used to represent the sum or difference of two integration constants.
3. The integration constant is always expressed as alone $C$ even if it uses trigonometric or logarithmic functions.
4. Only indefinite integrals make use of the integration constant; definite integrals do not.
5. The constant of integration, which is shown as a constant $C$, is multiplied or divided by the same value when the integral is multiplied or divided.
6. We use the same alphabet $C$ to represent the constant of integration despite multiple integration techniques and various expected integration constant values.

Hence, the constant of integration is explained