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.