Calculus Examples

$\frac{d P}{d x} = 3 x \sqrt{x^{2} + 9}$

Step 1

Rewrite the equation.

$d P = 3 x \sqrt{x^{2} + 9} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d P = \int 3 x \sqrt{x^{2} + 9} d x$

Step 2.2

Apply the constant rule.

$P + C_{1} = \int 3 x \sqrt{x^{2} + 9} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$P + C_{1} = 3 \int x \sqrt{x^{2} + 9} d x$

Step 2.3.2

Let $u = x^{2} + 9$ . 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 2.3.2.1

Let $u = x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{2} + 9$ .

$\frac{d}{d x} [x^{2} + 9]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 2.3.2.1.3

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

$2 x + \frac{d}{d x} [9]$

Step 2.3.2.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.2.2

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

$P + C_{1} = 3 \int \sqrt{u} \frac{1}{2} d u$

Step 2.3.3

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

$P + C_{1} = 3 \int \frac{\sqrt{u}}{2} d u$

Step 2.3.4

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

$P + C_{1} = 3 (\frac{1}{2} \int \sqrt{u} d u)$

Step 2.3.5

Combine fractions.

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Step 2.3.5.1

Combine $\frac{1}{2}$ and $3$ .

$P + C_{1} = \frac{3}{2} \int \sqrt{u} d u$

Step 2.3.5.2

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

$P + C_{1} = \frac{3}{2} \int u^{\frac{1}{2}} d u$

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Multiply $3$ by $2$ .

$P + C_{1} = \frac{6}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.3

Multiply $2$ by $3$ .

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

Step 2.3.7.2.4

Cancel the common factor of $6$ .

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Step 2.3.7.2.4.1

Cancel the common factor.

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

Step 2.3.7.2.4.2

Rewrite the expression.

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

Step 2.3.7.2.5

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

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

Step 2.3.8

Replace all occurrences of $u$ with $x^{2} + 9$ .

$P + C_{1} = {(x^{2} + 9)}^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$P = {(x^{2} + 9)}^{\frac{3}{2}} + K$