Calculus Examples

$\frac{d y}{d x} = \frac{\sqrt{x}}{x} i$

Step 1

Rewrite the equation.

$d y = \frac{\sqrt{x}}{x} i d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{\sqrt{x}}{x} i d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Combine $\frac{\sqrt{x}}{x}$ and $i$ .

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2.3.3.2.2

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.3.2.2.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 2.3.3.2.2.1.1

Raise $x$ to the power of $1$ .

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

Step 2.3.3.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.3.2.2.3

Combine the numerators over the common denominator.

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

Step 2.3.3.2.2.4

Subtract $1$ from $2$ .

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

Step 2.3.3.3

Apply basic rules of exponents.

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Step 2.3.3.3.1

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

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

Step 2.3.3.3.2

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

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Step 2.3.3.3.2.1

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

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

Step 2.3.3.3.2.2

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

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

Step 2.3.3.3.2.3

Move the negative in front of the fraction.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Move $2$ to the left of $i$ .

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

Step 2.4

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

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