Calculus Examples

$\frac{d y}{d x} = \sqrt{x} - \frac{2}{x}$

Step 1

Rewrite the equation.

$d y = (\sqrt{x} - \frac{2}{x}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int \sqrt{x} - \frac{2}{x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sqrt{x} - \frac{2}{x} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \sqrt{x} d x + \int - \frac{2}{x} d x$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y + C_{1} = \int x^{\frac{1}{2}} d x + \int - \frac{2}{x} d x$

Step 2.3.3

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} + \int - \frac{2}{x} d x$

Step 2.3.4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} - \int \frac{2}{x} d x$

Step 2.3.5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} - (2 \int \frac{1}{x} d x)$

Step 2.3.6

Multiply $2$ by $- 1$ .

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} - 2 \int \frac{1}{x} d x$

Step 2.3.7

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} - 2 (\ln (| x |) + C_{3})$

Step 2.3.8

Simplify.

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} - 2 \ln (| x |) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$y = \frac{2}{3} x^{\frac{3}{2}} - 2 \ln (| x |) + C$