Mathematics · Calculus
Power Rule Integral Calculator
Integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Increase power: 2 + 1 = 3.
- Divide coefficient: 3 ÷ 3 = 1.
- At x = 2: 1 × 2^3 = 8; append + C to the general antiderivative.
Understand Power-rule integral
One idea, three depths
Choose how deeply to explain Power-rule integral
Power-rule integral: Integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Power-rule integral to answer this question: integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value? Enter Coefficient a, Power n, Evaluation point x; the calculator shows Antiderivative coefficient. For example: ∫3x² dx = x³ + C; at x = 2 the nonconstant part equals 8. The answer tells you Antiderivative coefficient.
Age 15Explain it to a 15-year-oldConnect it to the formula
Integration reverses differentiation: increase the power by one and divide by that new power. The rule is ∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1. Its input values are Coefficient a, Power n, Evaluation point x, and the main result is Antiderivative coefficient. For example: ∫3x² dx = x³ + C; at x = 2 the nonconstant part equals 8.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated power-rule integral relation over the valid real-number domain stated below. The implemented relation is ∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1, evaluated from Coefficient a, Power n, Evaluation point x to produce Antiderivative coefficient. Integration reverses differentiation: increase the power by one and divide by that new power. The rule does not apply to n = −1, whose antiderivative is a ln|x| + C.
Inputs and valid domain
- Coefficient a must be a finite real number.
- Power n must be a finite real number.
- Evaluation point x must be a finite real number.
Important boundary: The rule does not apply to n = −1, whose antiderivative is a ln|x| + C.
The formula
∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1
How the calculator works through it
It substitutes Coefficient a, Power n, Evaluation point x into the formula and exposes every numerical step above. The main output is Antiderivative coefficient, accompanied by New power, Value excluding + C.
Read the result correctly
The Antiderivative coefficient is the direct answer to “integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
∫3x² dx = x³ + C; at x = 2 the nonconstant part equals 8.
Where this model stops being reliable
The rule does not apply to n = −1, whose antiderivative is a ln|x| + C.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Power-rule integral works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Power-rule integral uses ∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives as rates of change
Rates of change explain the local behaviour captured or approximated by Power-rule integral.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Power-rule integral to accumulated change, area and continuous totals.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Coefficient a, Power n, Evaluation point x.
- Evaluate the principal relationship: ∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1.
- Return Antiderivative coefficient and check the domain conditions described above.
Python
from math import *
def power_rule_integral(a, n, x) -> float:
return (a / (n + 1.0))
assert abs(power_rule_integral(3, 2, 2) - 1) < 1e-6 * max(1.0, abs(1))
C
#include <assert.h>
#include <math.h>
double power_rule_integral(double a, double n, double x) {
return (a / (n + 1.0));
}
int main(void) {
const double expected = 1;
const double actual = power_rule_integral(3, 2, 2);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double power_rule_integral(double a, double n, double x) {
return (a / (n + 1.0));
}
int main() {
constexpr double expected = 1;
const double actual = power_rule_integral(3, 2, 2);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double power_rule_integral(double a, double n, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global power_rule_integral
section .text
power_rule_integral:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-16]
addsd xmm0, [rbp-48]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
divsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = power_rule_integral(a, n, x)
result = (a / (n + 1.0));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, n_, x_] := (a / (n + 1.0));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Power Rule Integral Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/power-rule-integral
MLA 9
MW SysArc. “Power Rule Integral Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/power-rule-integral. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Power Rule Integral Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/calculus/power-rule-integral.
Harvard
MW SysArc (2026) ‘Power Rule Integral Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/power-rule-integral (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_power_rule_integral_2026,
author = {{MW SysArc}},
title = {Power Rule Integral Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/power-rule-integral},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Power Rule Integral Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/calculus/power-rule-integral
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Power-rule integral do?
Integrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
How does the Power-rule integral work?
The calculator applies ∫axⁿ dx = a/(n+1)xⁿ⁺¹ + C, for n ≠ −1. Integration reverses differentiation: increase the power by one and divide by that new power.
What can I learn from the Power-rule integral?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .