Calculus Examples

$(x^{2} + 3) \frac{d y}{d x} - x = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Apply the distributive property.

$x^{2} \frac{d y}{d x} + 3 \frac{d y}{d x} - x = 0$

Step 1.1.2

Add $x$ to both sides of the equation.

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

Step 1.1.3

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x} + 3 \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x}$ .

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

Step 1.1.3.2

Factor $\frac{d y}{d x}$ out of $3 \frac{d y}{d x}$ .

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

Step 1.1.3.3

Factor $\frac{d y}{d x}$ out of $\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot 3$ .

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

Cancel the common factor of $x^{2} + 3$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

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

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x^{2} + 3$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = x^{2} + 3$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x^{2} + 3$ .

$\frac{d}{d x} [x^{2} + 3]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [3]$

Step 2.3.1.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

$y + C_{1} = \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

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

$y + C_{1} = \int \frac{1}{u \cdot 2} d u$

Step 2.3.2.2

Move $2$ to the left of $u$ .

$y + C_{1} = \int \frac{1}{2 u} d u$

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = \frac{1}{2} \int \frac{1}{u} d u$

Step 2.3.4

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

$y + C_{1} = \frac{1}{2} (\ln (| u |) + C_{2})$

Step 2.3.5

Simplify.

$y + C_{1} = \frac{1}{2} \ln (| u |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $x^{2} + 3$ .

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

Step 2.4

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

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