# Quick Math Integral Calculus

## Quick Math Integral Calculus

1
##### 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.