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Integrals · Basic Antiderivatives

Power Rule for Integration

∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

The reverse of the power rule for derivatives. Increase the exponent by 1 and divide by the new exponent.

Conditions. n ≠ -1 (that case gives ln|x|).

Worked examples

Find ∫ x⁴ dx.
  1. x⁵/5 + C

Answer: x⁵/5 + C

Find ∫ 1/x³ dx.
  1. Rewrite as ∫ x⁻³ dx = x⁻²/(-2) + C = -1/(2x²) + C

Answer: -1/(2x²) + C

Find ∫ √x dx.
  1. ∫ x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3)x^(3/2) + C

Answer: (2/3)x^(3/2) + C

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