## Quick Math Integral Calculus

##### Enter expression in this form

Example: 3x^2 + 8x -12

\[ f(x) = x^3 + 2x^2 + x \]

### Integral Calculus in Calculus

Integral calculus is a branch of calculus focused on the concept of integration, which is the process of finding the area under a curve. Integration is essentially the reverse process of differentiation and is used to accumulate quantities, such as areas, volumes, and total values.

#### Definition

The definite integral of a function \( f(x) \) from \( a \) to \( b \) is defined as:

\[ \int_{a}^{b} f(x) \, dx \]

This represents the area under the curve of \( f(x) \) from \( x = a \) to \( x = b \).

The indefinite integral (or antiderivative) of a function \( f(x) \) is given by:

\[ \int f(x) \, dx = F(x) + C \]

where \( F(x) \) is the antiderivative of \( f(x) \) and \( C \) is the constant of integration.

#### Examples

Example 1: Indefinite Integral of a Polynomial Function

Function:

\[ f(x) = 3x^2 \]

Integral:

\[ \int 3x^2 \, dx = x^3 + C \]

The antiderivative of \( 3x^2 \) is \( x^3 \), plus the constant of integration \( C \).

#### Example 2: Definite Integral of a Polynomial Function

Function:

\[ f(x) = 2x \]

Integral from \( a = 1 \) to \( b = 3 \):

\[ \int_{1}^{3} 2x \, dx = \left[ x^2 \right]_{1}^{3} \]

\[ = 3^2 – 1^2 \]

\[ = 9 – 1 \]

\[ = 8 \]

The definite integral of \( 2x \) from 1 to 3 is 8.

#### Example 3: Indefinite Integral of a Trigonometric Function

Function:

\[ f(x) = \cos(x) \]

Integral:

\[ \int \cos(x) \, dx = \sin(x) + C \]

The antiderivative of \( \cos(x) \) is \( \sin(x) \), plus the constant of integration \( C \).

#### Example 4: Definite Integral of a Trigonometric Function

Function:

\[ f(x) = \sin(x) \]

Integral from \( a = 0 \) to \( b = \frac{\pi}{2} \):

\[ \int_{0}^{\frac{\pi}{2}} \sin(x) \, dx = \left[ -\cos(x) \right]_{0}^{\frac{\pi}{2} } \]

\[ = -\cos\left(\frac{\pi}{2}\right) – (-\cos(0)) \]

\[ = -0 + 1 \]

\[ = 1 \]

The definite integral of \( \sin(x) \) from 0 to \( \frac{\pi}{2} \) is 1.

#### Example 5: Indefinite Integral of an Exponential Function

Function:

\[ f(x) = e^x \]

Integral:

\[ \int e^x \, dx = e^x + C \]

The antiderivative of \( e^x \) is \( e^x \), plus the constant of integration \( C \).

Integral calculus is a fundamental tool in many fields for accumulating quantities and understanding the overall behavior of functions over intervals.