Calculus Examples

$\frac{d y}{d x} = \frac{1}{2 \sqrt{x}} + 3$ , $y (9) = 1$

Step 1

Rewrite the equation.

$d y = (\frac{1}{2 \sqrt{x}} + 3) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{1}{2 \sqrt{x}} + 3 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{1}{2 \sqrt{x}} + 3 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \frac{1}{2 \sqrt{x}} d x + \int 3 d x$

Step 2.3.2

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

$y + C_{1} = \frac{1}{2} \int \frac{1}{\sqrt{x}} d x + \int 3 d x$

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 $x^{\frac{1}{2}}$ .

$y + C_{1} = \frac{1}{2} \int \frac{1}{x^{\frac{1}{2}}} d x + \int 3 d x$

Step 2.3.3.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{1} = \frac{1}{2} \int x^{\frac{1}{2} \cdot - 1} d x + \int 3 d x$

Step 2.3.3.3.2

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

$y + C_{1} = \frac{1}{2} \int x^{\frac{- 1}{2}} d x + \int 3 d x$

Step 2.3.3.3.3

Move the negative in front of the fraction.

$y + C_{1} = \frac{1}{2} \int x^{- \frac{1}{2}} d x + \int 3 d x$

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

$y = 3 x + x^{\frac{1}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $9$ for $x$ and $1$ for $y$ in $y = 3 x + x^{\frac{1}{2}} + K$ .

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

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $3 \cdot 9 + 9^{\frac{1}{2}} + K = 1$ .

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

Step 4.2

Simplify $3 \cdot 9 + 9^{\frac{1}{2}} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $3$ by $9$ .

$27 + 9^{\frac{1}{2}} + K = 1$

Step 4.2.1.2

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

$27 + {(3^{2})}^{\frac{1}{2}} + K = 1$

Step 4.2.1.3

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

$27 + 3^{2 (\frac{1}{2})} + K = 1$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Cancel the common factor.

$27 + 3^{2 (\frac{1}{2})} + K = 1$

Step 4.2.1.4.2

Rewrite the expression.

$27 + 3^{1} + K = 1$

Step 4.2.1.5

Evaluate the exponent.

$27 + 3 + K = 1$

Step 4.2.2

Add $27$ and $3$ .

$30 + K = 1$

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 $30$ from both sides of the equation.

$K = 1 - 30$

Step 4.3.2

Subtract $30$ from $1$ .

$K = - 29$

Step 5

Substitute $- 29$ for $K$ in $y = 3 x + x^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $- 29$ for $K$ .

$y = 3 x + x^{\frac{1}{2}} - 29$