Integrals: the basics

The formula for ∫xⁿ dx

The indefinite integral of xnx^n is xn+1n+1+C\dfrac{x^{n+1}}{n+1}+C: raise the exponent by 1 and divide by the new exponent (n≠−1n\neq -1).

∫xn dx=xn+1n+1+C(n≠−1)\begin{gathered} \int x^n\,dx \\[4pt] =\frac{x^{n+1}}{n+1}+C\quad(n\neq -1) \end{gathered}

Differentiation brought the exponent down and reduced it by 1. Integration is the reverse: raise the exponent by 1 and divide by the new exponent.

∫x3 dx=x44+C\int x^3\,dx=\frac{x^4}{4}+C

Checking is easy: differentiate your answer and see that you get back the original.

Examples

  1. 01Integrate∫x2 dx\displaystyle \int x^{2} \, dxAnswerHide
    Answer13x3+C\displaystyle \frac{1}{3}x^{3} + C
  2. 02Integrate∫x5 dx\displaystyle \int x^{5} \, dxAnswerHide
    Answer16x6+C\displaystyle \frac{1}{6}x^{6} + C
  3. 03Integrate∫x dx\displaystyle \int x \, dxAnswerHide
    Answer12x2+C\displaystyle \frac{1}{2}x^{2} + C
  4. 04Integrate∫x3 dx\displaystyle \int x^{3} \, dxAnswerHide
    Answer14x4+C\displaystyle \frac{1}{4}x^{4} + C
  5. 05Integrate∫x4 dx\displaystyle \int x^{4} \, dxAnswerHide
    Answer15x5+C\displaystyle \frac{1}{5}x^{5} + C

FAQ

What about n=−1n=-1?
The denominator of the formula becomes 0, so it cannot be used. That case has its own formula: ∫1x dx=log⁡∣x∣+C\int \dfrac{1}{x}\,dx=\log|x|+C.

Published 2026-09-03 · Updated 2026-10-15