integrate sqrt(x) — step-by-step solution

Let's compute the integral:

\int \sqrt{x} d x

Use the integration formula for

\sqrt{x}

\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^{\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}

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

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

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