Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 4 - x^{2}$ . 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 $u = 4 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $4 - x^{2}$ .

$\frac{d}{d x} [4 - x^{2}]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $4 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$

Step 2.3.1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 2.3.1.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 2.3.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.3.2

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

$0 - (2 x)$

Step 2.3.1.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 2.3.1.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 2.3.1.2

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$y + C_{1} = \int \sqrt{u} (- \frac{1}{2}) d u$

Step 2.3.2.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

$y + C_{1} = - (\frac{1}{2} \int \sqrt{u} d u)$

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Multiply $3$ by $2$ .

$y + C_{1} = - \frac{2}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.3

Cancel the common factor of $2$ and $6$ .

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Step 2.3.7.2.3.1

Factor $2$ out of $2$ .

$y + C_{1} = - \frac{2 (1)}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.3.2

Cancel the common factors.

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Step 2.3.7.2.3.2.1

Factor $2$ out of $6$ .

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

Step 2.3.7.2.3.2.2

Cancel the common factor.

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

Step 2.3.7.2.3.2.3

Rewrite the expression.

$y + C_{1} = - \frac{1}{3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 2.4

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

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