integrate 1/sqrt(x) — step-by-step solution

Let's compute the integral:

\int \frac{1}{\sqrt{x}} d x

For the math expression:

\frac{1}{\sqrt{x}}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

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

Let's perform the differentiation:

\frac{d}{d x} (x^{- \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{d}{d x} (x^{\frac{- 1 + 2}{2}})

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

- 1 + 2

1

\frac{d}{d x} (\sqrt{x})

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d x} (\sqrt{x})

\frac{1}{2 \sqrt{x}}

\frac{1}{2 \sqrt{x}}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

2 \sqrt{x}

\frac{1}{\sqrt{x}}

, so

2 \sqrt{x}

is an antiderivative of

\frac{1}{\sqrt{x}}

\int \frac{1}{\sqrt{x}} d x

2 \sqrt{x}

2 \sqrt{x}

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

2 \sqrt{x} + C