Calculas

Definite Integral Calculator for axⁿ

Integral Result:

" + coefficientA + "x^{" + exponentN + "}

" + lowerLimitB + "

" + upperLimitC + "

" + integralValue.toFixed(6) + "

Understanding the Definite Integral of axⁿ

Calculus is a branch of mathematics focused on rates of change and accumulation. One of its fundamental concepts is the integral, which can be thought of as the reverse process of differentiation. While indefinite integrals yield a family of functions, definite integrals provide a specific numerical value, often representing quantities like area under a curve, volume, or total change.

What is a Definite Integral?

A definite integral calculates the net accumulation of a quantity over a specific interval. Geometrically, for a non-negative function, it represents the area between the function's graph and the x-axis over that interval. It is denoted by:

∫_b^c f(x) dx

Where:

• ∫ is the integral symbol.
• f(x) is the function being integrated (the integrand).
• dx indicates that the integration is with respect to the variable x.
• b is the lower limit of integration.
• c is the upper limit of integration.

Integrating the Power Function axⁿ

Our calculator focuses on a common type of function: the power function axn, where a is a constant coefficient and n is a constant exponent. The general rule for finding the indefinite integral of xn is:

∫ xⁿ dx = (1 / (n+1)) xⁿ⁺¹ + C, for n ≠ -1

When a coefficient a is present, it simply multiplies the result:

∫ axⁿ dx = (a / (n+1)) xⁿ⁺¹ + C, for n ≠ -1

For the special case where n = -1 (i.e., ax-1 or a/x), the integral is:

∫ a/x dx = a ln|x| + C

To find the definite integral from b to c, we evaluate the antiderivative at the upper limit c and subtract its value at the lower limit b. This is known as the Fundamental Theorem of Calculus:

∫_b^c f(x) dx = F(c) – F(b)

Where F(x) is the antiderivative of f(x).

How to Use This Calculator

This calculator simplifies the process of finding the definite integral for functions of the form axn. Follow these steps:

1. Coefficient (a): Enter the constant multiplier of your xn term. For example, if your function is 3x2, enter 3. If it's just x2, enter 1.
2. Exponent (n): Enter the power to which x is raised. For 3x2, enter 2. For 5/x (which is 5x-1), enter -1.
3. Lower Limit (b): Enter the starting point of your integration interval.
4. Upper Limit (c): Enter the ending point of your integration interval.
5. Click "Calculate Integral" to see the result.

Examples

Let's look at a few examples to illustrate the calculator's use:

Example 1: Area under a Parabola

Calculate the definite integral of 2x2 from x = 0 to x = 3.

• Coefficient (a): 2
• Exponent (n): 2
• Lower Limit (b): 0
• Upper Limit (c): 3

Calculation:
Antiderivative F(x) = (2 / (2+1))x²⁺¹ = (2/3)x³
F(3) – F(0) = (2/3)(3)³ – (2/3)(0)³ = (2/3)(27) – 0 = 18
Result: 18

Example 2: Integral of a Reciprocal Function

Calculate the definite integral of 5/x from x = 1 to x = 5.

• Coefficient (a): 5
• Exponent (n): -1
• Lower Limit (b): 1
• Upper Limit (c): 5

Calculation:
Antiderivative F(x) = 5 ln|x|
F(5) – F(1) = 5 ln(5) – 5 ln(1) = 5 ln(5) – 0 ≈ 5 * 1.6094 = 8.047
Result: Approximately 8.047

Example 3: Integral of a Linear Function

Calculate the definite integral of 4x from x = 1 to x = 2.

• Coefficient (a): 4
• Exponent (n): 1
• Lower Limit (b): 1
• Upper Limit (c): 2

Calculation:
Antiderivative F(x) = (4 / (1+1))x¹⁺¹ = 2x²
F(2) – F(1) = 2(2)² – 2(1)² = 2(4) – 2(1) = 8 – 2 = 6
Result: 6