AP Calculus AB and BC

Integral of x^2 sqrt(x): Rewrite as One Power

The integral of x squared times the square root of x is 2 times x to the seven halves, over 7, plus C. Combining the factors gives x to the five halves, and the power rule adds one to the exponent and divides.

∫x2x dx=2x7/27+C\int x^{2}\sqrt{x}\,dx = \frac{2x^{7/2}}{7} + C

Combine, then apply the power rule

x2x=x2⋅x1/2=x5/2x^{2}\sqrt{x} = x^{2}\cdot x^{1/2} = x^{5/2}
∫x5/2dx=x7/27/2+C=2x7/27+C\int x^{5/2}dx = \frac{x^{7/2}}{7/2} + C = \frac{2x^{7/2}}{7} + C

Dividing by a fraction

Adding one to 5/2 gives 7/2, and dividing by 7/2 is multiplying by 2/7. That step is where most of the arithmetic errors on fractional powers happen.

Common mistakes

  • Integrating the two factors separately and multiplying. There is no product rule for integrals.
  • Writing 7x7/22\frac{7x^{7/2}}{2}, inverting the fraction the wrong way.
  • Adding the exponents wrongly: 2+12=522 + \frac{1}{2} = \frac{5}{2}, not 32\frac{3}{2}.

Every answer on this page is machine checked

An automated test differentiates the antiderivative above and confirms it returns the integrand. A wrong sign or a missing factor fails the build, so it cannot reach you.

Frequently asked questions

What is the integral of x^2 sqrt(x)?

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

Why 2/7?

The new exponent is 72\frac{7}{2}, and dividing by 72\frac{7}{2} means multiplying by 27\frac{2}{7}.