EZ Math

Calculus · Integrals

How to Integrate a Square Root

Problem

∫x dx\int \sqrt{x}\,dx

Answer

23x3/2+C\frac{2}{3}x^{3/2} + C

A square root is a power with exponent 1/2, so the ordinary power rule for integrals handles it. Rewrite the root, then integrate.

Step-by-step solution

  1. Rewrite the root as a power.

    x=x1/2\sqrt{x} = x^{1/2}

    Why: The power rule needs an exponent.

  2. Add one to the exponent.

    12+1=32\frac{1}{2} + 1 = \frac{3}{2}

    Why: The integral rule raises the power by one.

  3. Divide by the new exponent and add C.

    x3/23/2+C=23x3/2+C\frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C

    Why: Dividing by 3/2 is the same as multiplying by 2/3.

Common mistakes

  • Integrating x^(1/2) to x^(1/2)/2. Add one to the exponent first.
  • Adding one to the coefficient instead of the exponent.
  • Dropping the constant C on an indefinite integral.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. ∫x3/2 dx\int x^{3/2}\,dx

    Show answer

    25x5/2+C\frac{2}{5}x^{5/2} + C

  2. ∫x1/3 dx\int x^{1/3}\,dx

    Show answer

    34x4/3+C\frac{3}{4}x^{4/3} + C

  3. ∫x−1/2 dx\int x^{-1/2}\,dx

    Show answer

    2x1/2+C2x^{1/2} + C

  4. ∫x5/2 dx\int x^{5/2}\,dx

    Show answer

    27x7/2+C\frac{2}{7}x^{7/2} + C

  5. ∫x2/3 dx\int x^{2/3}\,dx

    Show answer

    35x5/3+C\frac{3}{5}x^{5/3} + C