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Calculus basics

Power rule for integrals

To integrate a power of x, raise the power by one and divide by the new power.

∫xndx=xn+1n+1+C, n≠−1
Definite form (the calculator)∫abcxndx=cbn+1−an+1n+1
The n = −1 case∫1xdx=ln⁡|x|+C

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−1−0.50.511.522.533.542468101214
  • Integral from 0 to 3 ≈ 9 (shaded; parts below the axis count as negative).

What the symbols mean

I
the value of the definite integral ∫abcxndx
c
the coefficient
n
the power (not −1)
a
the lower limit
b
the upper limit
C
the constant of integration (indefinite integrals only)

When to use it

Use it to integrate polynomials term by term, and roots or reciprocals written as powers. For n = −1 the formula would divide by zero; the integral of 1/x is ln⁡|x|, which the calculator uses automatically. If xn blows up inside the interval (n ≤ −1 with 0 between the limits) the integral has no finite value, and the calculator says so.

Worked example

Find the integral of x² from 0 to 3 (c = 1, n = 2, a = 0, b = 3).

  1. Power rule for integrals: raise the power by one and divide by the new power.

    ∫abcxndx=cbn+1−an+1n+1

  2. Substitute and work it out:

    I=1·(32+1−02+1)2+1
    =1·(33−03)3
    =1·(27−0)3
    =1·273
    =273
    =9

  3. Numerical check: Simpson's rule with 400 strips gives 9.

Answer The integral is 9

Why it works

It is the power rule for derivatives run backwards: differentiating xn+1n+1 gives xn.

Common mistakes

  • Dividing by the old power instead of the new one: ∫x2dx=x3/3, not x3/2.
  • Using it for n = −1; ∫x−1dx=ln⁡|x|.
  • Forgetting + C on an indefinite integral.
  • Integrating across x = 0 when n ≤ −1: ∫−11x−2dx is not −2; it diverges.