Calculus Examples

$\frac{d y}{d x} = \frac{1}{9 \sqrt{x}}$ , $y (9) = 0$

Step 1

Rewrite the equation.

$d y = \frac{1}{9 \sqrt{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{1}{9 \sqrt{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{1}{9 \sqrt{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

$y + C_{1} = \frac{1}{9} \int \frac{1}{\sqrt{x}} d x$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

$y + C_{1} = \frac{1}{9} \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 2.3.2.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{1} = \frac{1}{9} \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.3.2.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.2.3.1

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

$y + C_{1} = \frac{1}{9} \int x^{\frac{1}{2} \cdot - 1} d x$

Step 2.3.2.3.2

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

$y + C_{1} = \frac{1}{9} \int x^{\frac{- 1}{2}} d x$

Step 2.3.2.3.3

Move the negative in front of the fraction.

$y + C_{1} = \frac{1}{9} \int x^{- \frac{1}{2}} d x$

Step 2.3.3

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

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

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $\frac{1}{9} (2 x^{\frac{1}{2}} + C_{2})$ as $\frac{1}{9} \cdot 2 x^{\frac{1}{2}} + C_{2}$ .

$y + C_{1} = \frac{1}{9} \cdot 2 x^{\frac{1}{2}} + C_{2}$

Step 2.3.4.2

Combine $\frac{1}{9}$ and $2$ .

$y + C_{1} = \frac{2}{9} x^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$y = \frac{2}{9} x^{\frac{1}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $9$ for $x$ and $0$ for $y$ in $y = \frac{2}{9} x^{\frac{1}{2}} + K$ .

$0 = \frac{2}{9} \cdot 9^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{2}{9} \cdot 9^{\frac{1}{2}} + K = 0$ .

$\frac{2}{9} \cdot 9^{\frac{1}{2}} + K = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{2}{9} \cdot {(3^{2})}^{\frac{1}{2}} + K = 0$

Step 4.2.2

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

$\frac{2}{9} \cdot 3^{2 (\frac{1}{2})} + K = 0$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{2}{9} \cdot 3^{2 (\frac{1}{2})} + K = 0$

Step 4.2.3.2

Rewrite the expression.

$\frac{2}{9} \cdot 3^{1} + K = 0$

Step 4.2.4

Evaluate the exponent.

$\frac{2}{9} \cdot 3 + K = 0$

Step 4.2.5

Cancel the common factor of $3$ .

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Step 4.2.5.1

Factor $3$ out of $9$ .

$\frac{2}{3 (3)} \cdot 3 + K = 0$

Step 4.2.5.2

Cancel the common factor.

$\frac{2}{3 \cdot 3} \cdot 3 + K = 0$

Step 4.2.5.3

Rewrite the expression.

$\frac{2}{3} + K = 0$

Step 4.3

Subtract $\frac{2}{3}$ from both sides of the equation.

$K = - \frac{2}{3}$

Step 5

Substitute $- \frac{2}{3}$ for $K$ in $y = \frac{2}{9} x^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $- \frac{2}{3}$ for $K$ .

$y = \frac{2}{9} x^{\frac{1}{2}} - \frac{2}{3}$

Step 5.2

Combine $\frac{2}{9}$ and $x^{\frac{1}{2}}$ .

$y = \frac{2 x^{\frac{1}{2}}}{9} - \frac{2}{3}$