integral of (-2sqrt(x)) — step-by-step solution

Let's compute the integral:

\int (- 2 \sqrt{x}) d x

We integrate

- 2 \sqrt{x}

by applying:

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

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int (- 2 \sqrt{x}) d x

- 2 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

- 2 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

ⓘ Multiplying a fraction by a number multiplies its numerator:

- 2 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

- \frac{2 x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}

■To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{2} + 1

\frac{1 + 2}{2}

- \frac{2 x^{\frac{1 + 2}{2}}}{\frac{1 + 2}{2}}

ⓘ The numerator and the denominator are computed before the division:

1 + 2

3

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

ⓘ Dividing is multiplying by the reciprocal:

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

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

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

ⅰ

2 \cdot \frac{2}{3}

\frac{4}{3}

- \frac{4}{3} \cdot x^{\frac{3}{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

- \frac{4}{3} \cdot x^{\frac{3}{2}}

- \frac{4 x^{\frac{3}{2}}}{3}

- \frac{4 x^{\frac{3}{2}}}{3}

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

- \frac{4 x^{\frac{3}{2}}}{3} + C