integrate sqrt(2x) — step-by-step solution

1) Let's integrate:

\int(\sqrt{2 x})dx

For

\sqrt{2 x}

, we use integration by parts:

\int(u1 \cdot v2')dx = u1 \cdot v2 + \int(u1' \cdot v2)dx

, with u1 =

\sqrt{2 x}

and v2' =

1

2) Let's perform the differentiation u1:

{(\sqrt{2 x})}^{'}

We differentiate using:

{(\sqrt{f})}^{'} = {(f)}^{'} \cdot \frac{1}{2 \sqrt{f}}

(where f =

2 x

)

{(2 x)}^{'} \cdot \frac{1}{2 \cdot \sqrt{2 x}}

\frac{2}{2 \cdot \sqrt{2 x}}

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

Thus,

u1'

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

3) Let's compute the integral v2':

\int(1)dx

x

4) Let's evaluate

u1' \cdot v2

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

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

\frac{x}{\sqrt{2} \cdot \sqrt{x}}

We apply simplification rules, we can replace the fraction:

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

with the new expression:

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

\frac{x^{(1 - \frac{1}{2})}}{\sqrt{2}}

\frac{x^{(\frac{2 - 1}{2})}}{\sqrt{2}}

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

The conventional preference is to have numerical square roots in the numerator rather than in the denominator:

\frac{\sqrt{x} \cdot \sqrt{2}}{2}

5) Let's compute the integral

u1 \cdot v2 + \int(u1' \cdot v2)dx

\sqrt{2 x} \cdot x - \int(\frac{\sqrt{x} \cdot \sqrt{2}}{2})dx

Use the integration formula for

\frac{\sqrt{x} \cdot \sqrt{2}}{2}

\int(x^{n})dx = \frac{x^{(n + 1)}}{n + 1}

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

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

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

\sqrt{2 x} \cdot x - \frac{\sqrt{2} \cdot x^{\frac{3}{2}}}{\frac{6}{2}}

\sqrt{2 x} \cdot x - \sqrt{2} \cdot x^{\frac{3}{2}} \cdot \frac{1}{3}

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

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

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

▷Answer: $- \frac{x^{\frac{3}{2}} \cdot \sqrt{2}}{3} + x \cdot \sqrt{2 x} + C$ , with C being any constant