Integration of constant term

Integrating a constant is a fundamental concept in calculus. When you integrate a constant ( c ) with respect to a variable ( x ), the result is a linear function in ( x ). The integration is performed as follows:

$$ \int c \, dx = c \cdot x + C $$

Here, ( C ) represents the constant of integration. This constant is added because when you differentiate a constant, the result is zero, and thus any constant could have been part of the original function before differentiation.

For example, if you integrate the constant 5 with respect to ( x ), you get:

$$ \int 5 \, dx = 5x + C $$

This means that the integral of 5 with respect to ( x ) is a linear function ( 5x ) plus an arbitrary constant ( C ).

Geometrical Meaning of Integration of constant

Integration of constant 5 represent family of curves as show in below figure .Each of which can be obtained by the shifting any one of the curves parallel to itself

Examples

1. Integration of 2:
   $$ \int 2 \, dx = 2x + C_1 $$
2. Integration of 3:
   $$ \int 3 \, dx = 3x + C_2 $$
3. Integration of 4:
   $$ \int 4 \, dx = 4x + C_3 $$
4. Integration of 6:
   $$ \int 6 \, dx = 6x + C_4 $$
5. Integration of ( \pi ) (Pi is a constant approximately equal to 3.14159):
   $$ \int \pi \, dx = \pi x + C_5 $$
6. Integration of ( e ) (e is Euler’s number, approximately equal to 2.71828):
   $$ \int e \, dx = ex + C_6 $$
7. Integration of 0:
   $$ \int 0 \, dx = 0x + C_7 = C_7 $$

In each case, the ( C_i ) represents a unique constant of integration for that particular integral. The integral of a constant is a linear function in ( x ), with the slope equal to the constant being integrated.

Definite Integral of constant value

Definite integral will be represent as

$\int _{a}^{b} c \, dx $

To solve this, we use the antiderivative of c, which is cx. So, the integral becomes:

\[
\left[ cx \right]_{a}^{b}
\]

Now, we evaluate this expression at the upper and lower limits of the integral:

\[
= cb -ca
\]

So, the definite integral of \(c \) from \(a\) to \(b \) is \(c(b-a)\).

This result has a geometric interpretation: it represents the area under the curve of (y = c) from (a) to (b), which corresponds to area of the rectangle