Calculus Examples

solve the differential equation of $\frac{d y}{d x} = \frac{2}{x^{2} + 4}$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Reorder $x^{2}$ and $4$ .

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

Step 2.3.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.3

The integral of $\frac{1}{2^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{2} \arctan (\frac{x}{2}) + C_{2}$ .

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

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

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

Step 2.3.4.2

Rewrite $2 (\frac{\arctan (\frac{x}{2})}{2} + C_{2})$ as $2 (\frac{1}{2}) \arctan (\frac{1}{2} x) + C_{2}$ .

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

Step 2.3.4.3

Simplify.

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Step 2.3.4.3.1

Combine $2$ and $\frac{1}{2}$ .

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

Step 2.3.4.3.2

Cancel the common factor of $2$ .

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Step 2.3.4.3.2.1

Cancel the common factor.

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

Step 2.3.4.3.2.2

Rewrite the expression.

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

Step 2.3.4.3.3

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.3.4.3.4

Multiply $\arctan (\frac{x}{2})$ by $1$ .

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

Step 2.3.4.4

Reorder terms.

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

Step 2.4

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

$y = \arctan (\frac{1}{2} x) + K$