Calculus Examples

$x \frac{d y}{d x} = 2 - 4 x^{3}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $x \frac{d y}{d x} = 2 - 4 x^{3}$ by $x$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $x \frac{d y}{d x} = 2 - 4 x^{3}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} = \frac{2}{x} + \frac{- 4 x^{3}}{x}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} = \frac{2}{x} + \frac{- 4 x^{3}}{x}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{2}{x} + \frac{- 4 x^{3}}{x}$

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 1.1.3.1.1

Factor $x$ out of $- 4 x^{3}$ .

$\frac{d y}{d x} = \frac{2}{x} + \frac{x (- 4 x^{2})}{x}$

Step 1.1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.3.1.2.1

Raise $x$ to the power of $1$ .

$\frac{d y}{d x} = \frac{2}{x} + \frac{x (- 4 x^{2})}{x^{1}}$

Step 1.1.3.1.2.2

Factor $x$ out of $x^{1}$ .

$\frac{d y}{d x} = \frac{2}{x} + \frac{x (- 4 x^{2})}{x \cdot 1}$

Step 1.1.3.1.2.3

Cancel the common factor.

$\frac{d y}{d x} = \frac{2}{x} + \frac{x (- 4 x^{2})}{x \cdot 1}$

Step 1.1.3.1.2.4

Rewrite the expression.

$\frac{d y}{d x} = \frac{2}{x} + \frac{- 4 x^{2}}{1}$

Step 1.1.3.1.2.5

Divide $- 4 x^{2}$ by $1$ .

$\frac{d y}{d x} = \frac{2}{x} - 4 x^{2}$

Step 1.2

Rewrite the equation.

$d y = (\frac{2}{x} - 4 x^{2}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{2}{x} - 4 x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{2}{x} - 4 x^{2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \frac{2}{x} d x + \int - 4 x^{2} d x$

Step 2.3.2

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

$y + C_{1} = 2 \int \frac{1}{x} d x + \int - 4 x^{2} d x$

Step 2.3.3

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

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

Step 2.3.4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Simplify.

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

Step 2.3.6.2

Simplify.

Tap for more steps...

Step 2.3.6.2.1

Combine $- 4$ and $\frac{1}{3}$ .

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

Step 2.3.6.2.2

Move the negative in front of the fraction.

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

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