integral of x-sqrt(x) — step-by-step solution

Let's compute the integral:

\int (x - \sqrt{x}) d x

ⅰ

\int (x - \sqrt{x}) d x

\int x d x - \int \sqrt{x} d x

\int x d x - \int \sqrt{x} d x

We integrate

\sqrt{x}

by applying:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

- \int \sqrt{x} d x

- \frac{x^{\frac{3}{2}}}{\frac{3}{2}}

\frac{x^{2}}{2} - \frac{x^{\frac{3}{2}}}{\frac{3}{2}}

ⓘ Dividing is multiplying by the reciprocal:

- \frac{x^{\frac{3}{2}}}{\frac{3}{2}}

- x^{\frac{3}{2}} \cdot \frac{2}{3}

\frac{x^{2}}{2} - x^{\frac{3}{2}} \cdot \frac{2}{3}

ⓘ Multiplying a fraction by a term multiplies its numerator:

x^{\frac{3}{2}} \cdot \frac{2}{3}

\frac{x^{\frac{3}{2}} \cdot 2}{3}

\frac{x^{2}}{2} - \frac{x^{\frac{3}{2}} \cdot 2}{3}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{x^{2}}{2} - \frac{2 x^{\frac{3}{2}}}{3} + C