Calculus Examples

$\frac{d y}{d x} = - \frac{1}{8 \sqrt{x}}$ , $y (25) = 0$

Step 1

Rewrite the equation.

$d y = - \frac{1}{8 \sqrt{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{1}{8 \sqrt{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{1}{8 \sqrt{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = - \int \frac{1}{8 \sqrt{x}} d x$

Step 2.3.2

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$y + C_{1} = - (\frac{1}{8} \int \frac{1}{\sqrt{x}} d x)$

Step 2.3.3

Apply basic rules of exponents.

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Step 2.3.3.1

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

$y + C_{1} = - \frac{1}{8} \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 2.3.3.2

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

$y + C_{1} = - \frac{1}{8} \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.3.3.3

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

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Step 2.3.3.3.1

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

$y + C_{1} = - \frac{1}{8} \int x^{\frac{1}{2} \cdot - 1} d x$

Step 2.3.3.3.2

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

$y + C_{1} = - \frac{1}{8} \int x^{\frac{- 1}{2}} d x$

Step 2.3.3.3.3

Move the negative in front of the fraction.

$y + C_{1} = - \frac{1}{8} \int x^{- \frac{1}{2}} d x$

Step 2.3.4

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

$y + C_{1} = - \frac{1}{8} (2 x^{\frac{1}{2}} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Rewrite $- \frac{1}{8} (2 x^{\frac{1}{2}} + C_{2})$ as $- \frac{1}{8} \cdot 2 x^{\frac{1}{2}} + C_{2}$ .

$y + C_{1} = - \frac{1}{8} \cdot 2 x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $2$ by $- 1$ .

$y + C_{1} = - 2 (\frac{1}{8}) x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.2

Combine $- 2$ and $\frac{1}{8}$ .

$y + C_{1} = \frac{- 2}{8} x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.3

Cancel the common factor of $- 2$ and $8$ .

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Step 2.3.5.2.3.1

Factor $2$ out of $- 2$ .

$y + C_{1} = \frac{2 (- 1)}{8} x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.3.2

Cancel the common factors.

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Step 2.3.5.2.3.2.1

Factor $2$ out of $8$ .

$y + C_{1} = \frac{2 \cdot - 1}{2 \cdot 4} x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.3.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 1}{2 \cdot 4} x^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.3.2.3

Rewrite the expression.

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

Step 2.3.5.2.4

Move the negative in front of the fraction.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $25$ for $x$ and $0$ for $y$ in $y = - \frac{1}{4} x^{\frac{1}{2}} + K$ .

$0 = - \frac{1}{4} \cdot 25^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{4} \cdot 25^{\frac{1}{2}} + K = 0$ .

$- \frac{1}{4} \cdot 25^{\frac{1}{2}} + K = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite $25$ as $5^{2}$ .

$- \frac{1}{4} \cdot {(5^{2})}^{\frac{1}{2}} + K = 0$

Step 4.2.2

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

$- \frac{1}{4} \cdot 5^{2 (\frac{1}{2})} + K = 0$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$- \frac{1}{4} \cdot 5^{2 (\frac{1}{2})} + K = 0$

Step 4.2.3.2

Rewrite the expression.

$- \frac{1}{4} \cdot 5^{1} + K = 0$

Step 4.2.4

Evaluate the exponent.

$- \frac{1}{4} \cdot 5 + K = 0$

Step 4.2.5

Multiply $- \frac{1}{4} \cdot 5$ .

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Step 4.2.5.1

Multiply $5$ by $- 1$ .

$- 5 (\frac{1}{4}) + K = 0$

Step 4.2.5.2

Combine $- 5$ and $\frac{1}{4}$ .

$\frac{- 5}{4} + K = 0$

Step 4.2.6

Move the negative in front of the fraction.

$- \frac{5}{4} + K = 0$

Step 4.3

Add $\frac{5}{4}$ to both sides of the equation.

$K = \frac{5}{4}$

Step 5

Substitute $\frac{5}{4}$ for $K$ in $y = - \frac{1}{4} x^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $\frac{5}{4}$ for $K$ .

$y = - \frac{1}{4} x^{\frac{1}{2}} + \frac{5}{4}$

Step 5.2

Combine $x^{\frac{1}{2}}$ and $\frac{1}{4}$ .

$y = - \frac{x^{\frac{1}{2}}}{4} + \frac{5}{4}$