Calculus Examples

$\frac{d f}{d x} = 45 x \sqrt{9 x^{2} + 4}$ , $f (0) = 9$

Step 1

Rewrite the equation.

$d f = 45 x \sqrt{9 x^{2} + 4} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d f = \int 45 x \sqrt{9 x^{2} + 4} d x$

Step 2.2

Apply the constant rule.

$f + C_{1} = \int 45 x \sqrt{9 x^{2} + 4} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $45$ is constant with respect to $x$ , move $45$ out of the integral.

$f + C_{1} = 45 \int x \sqrt{9 x^{2} + 4} d x$

Step 2.3.2

Let $u = 9 x^{2} + 4$ . Then $d u = 18 x d x$ , so $\frac{1}{18} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 9 x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $9 x^{2} + 4$ .

$\frac{d}{d x} [9 x^{2} + 4]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $9 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [4]$

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 2.3.2.1.3.2

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

$9 (2 x) + \frac{d}{d x} [4]$

Step 2.3.2.1.3.3

Multiply $2$ by $9$ .

$18 x + \frac{d}{d x} [4]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

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

$18 x + 0$

Step 2.3.2.1.4.2

Add $18 x$ and $0$ .

$18 x$

Step 2.3.2.2

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

$f + C_{1} = 45 \int \sqrt{u} \frac{1}{18} d u$

Step 2.3.3

Combine $\sqrt{u}$ and $\frac{1}{18}$ .

$f + C_{1} = 45 \int \frac{\sqrt{u}}{18} d u$

Step 2.3.4

Since $\frac{1}{18}$ is constant with respect to $u$ , move $\frac{1}{18}$ out of the integral.

$f + C_{1} = 45 (\frac{1}{18} \int \sqrt{u} d u)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Simplify.

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Step 2.3.5.1.1

Combine $\frac{1}{18}$ and $45$ .

$f + C_{1} = \frac{45}{18} \int \sqrt{u} d u$

Step 2.3.5.1.2

Cancel the common factor of $45$ and $18$ .

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Step 2.3.5.1.2.1

Factor $9$ out of $45$ .

$f + C_{1} = \frac{9 (5)}{18} \int \sqrt{u} d u$

Step 2.3.5.1.2.2

Cancel the common factors.

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Step 2.3.5.1.2.2.1

Factor $9$ out of $18$ .

$f + C_{1} = \frac{9 \cdot 5}{9 \cdot 2} \int \sqrt{u} d u$

Step 2.3.5.1.2.2.2

Cancel the common factor.

$f + C_{1} = \frac{9 \cdot 5}{9 \cdot 2} \int \sqrt{u} d u$

Step 2.3.5.1.2.2.3

Rewrite the expression.

$f + C_{1} = \frac{5}{2} \int \sqrt{u} d u$

Step 2.3.5.2

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

$f + C_{1} = \frac{5}{2} \int u^{\frac{1}{2}} d u$

Step 2.3.6

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

$f + C_{1} = \frac{5}{2} (\frac{2}{3} u^{\frac{3}{2}} + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite $\frac{5}{2} (\frac{2}{3} u^{\frac{3}{2}} + C_{2})$ as $\frac{5}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C_{2}$ .

$f + C_{1} = \frac{5}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $\frac{5}{2}$ by $\frac{2}{3}$ .

$f + C_{1} = \frac{5 \cdot 2}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.2

Multiply $5$ by $2$ .

$f + C_{1} = \frac{10}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.3

Multiply $2$ by $3$ .

$f + C_{1} = \frac{10}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.4

Cancel the common factor of $10$ and $6$ .

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Step 2.3.7.2.4.1

Factor $2$ out of $10$ .

$f + C_{1} = \frac{2 (5)}{6} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.4.2

Cancel the common factors.

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Step 2.3.7.2.4.2.1

Factor $2$ out of $6$ .

$f + C_{1} = \frac{2 \cdot 5}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.4.2.2

Cancel the common factor.

$f + C_{1} = \frac{2 \cdot 5}{2 \cdot 3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.7.2.4.2.3

Rewrite the expression.

$f + C_{1} = \frac{5}{3} u^{\frac{3}{2}} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $9 x^{2} + 4$ .

$f + C_{1} = \frac{5}{3} {(9 x^{2} + 4)}^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$f = \frac{5}{3} {(9 x^{2} + 4)}^{\frac{3}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $9$ for $f$ in $f = \frac{5}{3} {(9 x^{2} + 4)}^{\frac{3}{2}} + K$ .

$9 = \frac{5}{3} {(9 \cdot 0^{2} + 4)}^{\frac{3}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{5}{3} \cdot {(9 \cdot 0^{2} + 4)}^{\frac{3}{2}} + K = 9$ .

$\frac{5}{3} \cdot {(9 \cdot 0^{2} + 4)}^{\frac{3}{2}} + K = 9$

Step 4.2

Simplify each term.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{5}{3} \cdot {(9 \cdot 0 + 4)}^{\frac{3}{2}} + K = 9$

Step 4.2.1.2

Multiply $9$ by $0$ .

$\frac{5}{3} \cdot {(0 + 4)}^{\frac{3}{2}} + K = 9$

Step 4.2.2

Add $0$ and $4$ .

$\frac{5}{3} \cdot 4^{\frac{3}{2}} + K = 9$

Step 4.2.3

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

$\frac{5}{3} \cdot {(2^{2})}^{\frac{3}{2}} + K = 9$

Step 4.2.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{3} \cdot 2^{2 (\frac{3}{2})} + K = 9$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$\frac{5}{3} \cdot 2^{2 (\frac{3}{2})} + K = 9$

Step 4.2.5.2

Rewrite the expression.

$\frac{5}{3} \cdot 2^{3} + K = 9$

Step 4.2.6

Raise $2$ to the power of $3$ .

$\frac{5}{3} \cdot 8 + K = 9$

Step 4.2.7

Multiply $\frac{5}{3} \cdot 8$ .

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Step 4.2.7.1

Combine $\frac{5}{3}$ and $8$ .

$\frac{5 \cdot 8}{3} + K = 9$

Step 4.2.7.2

Multiply $5$ by $8$ .

$\frac{40}{3} + K = 9$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $\frac{40}{3}$ from both sides of the equation.

$K = 9 - \frac{40}{3}$

Step 4.3.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$K = 9 \cdot \frac{3}{3} - \frac{40}{3}$

Step 4.3.3

Combine $9$ and $\frac{3}{3}$ .

$K = \frac{9 \cdot 3}{3} - \frac{40}{3}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{9 \cdot 3 - 40}{3}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $9$ by $3$ .

$K = \frac{27 - 40}{3}$

Step 4.3.5.2

Subtract $40$ from $27$ .

$K = \frac{- 13}{3}$

Step 4.3.6

Move the negative in front of the fraction.

$K = - \frac{13}{3}$

Step 5

Substitute $- \frac{13}{3}$ for $K$ in $f = \frac{5}{3} {(9 x^{2} + 4)}^{\frac{3}{2}} + K$ and simplify.

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Step 5.1

Substitute $- \frac{13}{3}$ for $K$ .

$f = \frac{5}{3} {(9 x^{2} + 4)}^{\frac{3}{2}} - \frac{13}{3}$

Step 5.2

Combine $\frac{5}{3}$ and ${(9 x^{2} + 4)}^{\frac{3}{2}}$ .

$f = \frac{5 {(9 x^{2} + 4)}^{\frac{3}{2}}}{3} - \frac{13}{3}$

Step 5.3

Combine the numerators over the common denominator.

$f = \frac{5 {(9 x^{2} + 4)}^{\frac{3}{2}} - 13}{3}$