formula.expert

Definite Integral of a Power Function

Also known as integral of x^n · definite integral of cx^n · power rule for integrals · reverse power rule · area under y = x^n · integrate x squared · antiderivative of x^n

I=∫abc xn dx=c(bn+1−an+1)n+1,n≠−1I = \int_{a}^{b} c\,x^{n}\,dx = \frac{c\left(b^{n+1} - a^{n+1}\right)}{n+1}, \quad n \neq -1

Units aren’t used in this calculation: every value is a plain number.

Worked example: ∫ 2x² dx from 1 to 3 → 52/3. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

See also
Power Rule Derivative: The rule this one undoes: differentiate c·xⁿ⁺¹/(n + 1) and c·xⁿ comes back.Average Value of a Function: What to do with the integral next: divide by b − a for the average height of the curve.Disc Method Volume for y = c·xⁿ: The same integral squared and spun about the axis: the volume of the solid the curve sweeps.Logarithm of a Quotient: The n = −1 case this page refuses: ∫ c/x dx from a to b is c·ln(b/a), a logarithm of a quotient.
Learning zone

Definite Integral of a Power Function explained

The definite integral of c·xⁿ from a to b undoes the power rule: raise the exponent by one, divide by the new exponent, and take the difference between the two limits. Worked example: the integral of 2x² from 1 to 3. The antiderivative is 2x³/3, which is 18 at x = 3 and 2/3 at x = 1, so the integral is 18 − 0.667 = 17.333. Fermat and Cavalieri had found the area under xⁿ by the 1640s; the fundamental theorem of calculus, a generation later, explained why the answer is an antiderivative read at two points.

The classic mistake is to read the integral as the area when part of the curve lies below the axis. The integral is a signed area, with the part below counted negative, so the integral of x³ from −1 to 1 is 0, even though the region between the curve and the axis has a total area of 0.5. For the area itself, split the interval where the curve crosses zero and add the sizes of the pieces. The other trap is n = −1, where the rule would divide by zero: 1/x integrates to ln|x|, which is why this page refuses it. The same integral is the work stored in a spring, the integral of kx from 0 to x, which is ½kx², and the distance covered by anything whose speed grows as a power of time.

Definite Integral of a Power Function formula

I=∫abc xn dx=c(bn+1−an+1)n+1,n≠−1I = \int_{a}^{b} c\,x^{n}\,dx = \frac{c\left(b^{n+1} - a^{n+1}\right)}{n+1}, \quad n \neq -1
Where
  • II= Value of the definite integral I
  • cc= Coefficient c
  • nn= Exponent n
  • aa= Lower limit a
  • bb= Upper limit b

Missing one of these? Work it out first, then come back

Revised 16 October 2026
Cite this page

formula.expert. (2026, October 16). Definite Integral of a Power Function Formula & Calculator. https://formula.expert/solvers/definite-integral-power

formula.experthttps://formula.expert/solvers/definite-integral-power