Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.3.1.1

Factor $x$ out of $x^{2}$ .

$\frac{d y}{d x} = \frac{x \cdot x}{3 x}$

Step 1.1.3.1.2

Cancel the common factors.

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Step 1.1.3.1.2.1

Factor $x$ out of $3 x$ .

$\frac{d y}{d x} = \frac{x \cdot x}{x \cdot 3}$

Step 1.1.3.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x \cdot x}{x \cdot 3}$

Step 1.1.3.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{x}{3}$

Step 1.2

Rewrite the equation.

$d y = \frac{x}{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{x}{3} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{1}{3} (\frac{1}{2} x^{2} + C_{2})$ as $\frac{1}{3} \cdot \frac{1}{2} x^{2} + C_{2}$ .

$y + C_{1} = \frac{1}{3} \cdot \frac{1}{2} x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$y + C_{1} = \frac{1}{3 \cdot 2} x^{2} + C_{2}$

Step 2.3.3.2.2

Multiply $3$ by $2$ .

$y + C_{1} = \frac{1}{6} x^{2} + C_{2}$

Step 2.4

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

$y = \frac{1}{6} x^{2} + K$