MathematicsCalculusA-Level

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Integral of x^n

The power rule for integration.

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\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

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Core idea

Overview

The Power Rule for integration provides a systematic way to find the antiderivative of a variable raised to a constant power. It dictates that the integral is found by increasing the exponent by one and dividing the expression by this new exponent value.

When to use: Use this rule when integrating power functions of the form xⁿ where n is any real number. Note that this specific formula applies only when the exponent n is not equal to -1, as that case requires a logarithmic solution.

Why it matters: This formula is the cornerstone of calculus used to calculate areas under curves, work done by variable forces, and moments of inertia. It allows engineers and scientists to move from rate-of-change models back to total accumulation models.

Symbols

Variables

I = Integral Value, x = x Value, n = Power

I

Integral Value

(i g n or in g C)

x

x Value

V a r iab l e

n

Power

V a r iab l e

Walkthrough

Derivation

Formula: Integral of x^n (Power Rule for Integration)

Integration reverses differentiation. The power rule for integration increases the power by 1 and divides by the new power.

n is a real number.
n $\neq =$ -1.

State the Rule:

Add 1 to the power, divide by the new power, and include the constant of integration C.

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Check by Differentiation:

Differentiating returns the original integrand, confirming the rule.

\frac{d}{d x} (\frac{x ^{n + 1}}{n + 1} + C) = x^{n}

Result

\frac{d}{d x} (\frac{x ^{n + 1}}{n + 1} + C) = x^{n}

Source: Standard curriculum — A-Level Pure Mathematics

Visual intuition

Graph

The graph of the integral value (I) against the independent variable (x) forms a power function curve where the exponent is n+1. As the independent variable increases, the curve grows at an accelerating rate for n > 0, reflecting the polynomial nature of the resulting expression.

Graph type: power_law

Why it behaves this way

Intuition

The integral represents the total area accumulated under the curve of the function y = xn by summing up an infinite number of infinitesimally thin vertical rectangles, each with height xn and width dx.

The independent variable of the function being integrated.

Represents the quantity along which accumulation is measured, such as position, time, or length.

The constant exponent of the independent variable.

Dictates the curvature or rate of change of the function xn, influencing how quickly the accumulated value grows or shrinks.

An infinitesimal increment of the independent variable x.

Represents the 'width' of an infinitely narrow slice, whose 'height' is xn, contributing to the total sum.

\int

The integral operator, signifying the process of antidifferentiation or summation.

Symbolizes the act of summing up an infinite number of infinitesimal contributions (xn dx) to find the total accumulated quantity or the net change.

The constant of integration.

Accounts for the unknown initial value or 'starting point' of the accumulated quantity, which is lost when a function is differentiated.

Signs and relationships

n+1 (in exponent): The exponent increases by one because integration is the inverse operation of differentiation, where the exponent decreases by one.
n+1 (in denominator): Division by the new exponent n+1 cancels out the factor that would appear if the result x^(n+1) were differentiated, ensuring the correct antiderivative.
+C: The constant C is added because the derivative of any constant is zero, meaning there is an arbitrary constant term in the original function that is recovered during indefinite integration.

Free study cues

Insight

Canonical usage

This equation is used to determine the antiderivative of a power function, where the dimension of the result is consistently one higher than the dimension of the original function's variable.

Common confusion

A common mistake is not correctly tracking how the units of the variable `x` propagate through the integration, leading to incorrect units for the antiderivative or the constant of integration `C`.

Unit systems

$x$ Varies by context (e.g., m, s, kg) · The dimension of the variable `x` determines the base dimension for the integrated result.

$d x$ Same as x · The differential `dx` carries the same dimension as the variable `x`.

$\int x^{n} d x$ Unit of x raised to the power (n+1) · The dimension of the integral result is the dimension of `x` raised to the power `n+1`.

$C$ Unit of x raised to the power (n+1) · The constant of integration `C` must have the same dimension as the integrated term `x^(n+1)/(n+1)`.

One free problem

Practice Problem

Find the value of the integral I = ∫ xⁿ dx given n = 2 and x = 3, assuming the constant of integration C is 0.

x Value3

Power2

Solve for: $I$

Hint: The integrated form is x³ / 3.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Finding position from velocity.

Study smarter

Tips

Always add the constant of integration C when performing indefinite integrals.
Check if the exponent is -1 before proceeding to avoid division by zero.
Convert radical signs or fractions into exponents (e.g., √x to $x^{0}$ .5) before integrating.
Verify your result by differentiating it; you should return to the original function.

Avoid these traps

Common Mistakes

Decreasing power.
Using for n=-1 (use ln).

Common questions

Frequently Asked Questions

Integration reverses differentiation. The power rule for integration increases the power by 1 and divides by the new power.

Use this rule when integrating power functions of the form xⁿ where n is any real number. Note that this specific formula applies only when the exponent n is not equal to -1, as that case requires a logarithmic solution.

This formula is the cornerstone of calculus used to calculate areas under curves, work done by variable forces, and moments of inertia. It allows engineers and scientists to move from rate-of-change models back to total accumulation models.

Decreasing power. Using for n=-1 (use ln).