Calculus Examples

$\frac{d p}{d x} = \sqrt[23]{\frac{P}{x}}$

Step 1

Rewrite the equation.

$d p = \sqrt[23]{\frac{P}{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d p = \int \sqrt[23]{\frac{P}{x}} d x$

Step 2.2

Apply the constant rule.

$p + C_{1} = \int \sqrt[23]{\frac{P}{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $\sqrt[23]{\frac{P}{x}}$ as $\frac{\sqrt[23]{P}}{\sqrt[23]{x}}$ .

$p + C_{1} = \int \frac{\sqrt[23]{P}}{\sqrt[23]{x}} d x$

Step 2.3.2

Since $\sqrt[23]{P}$ is constant with respect to $x$ , move $\sqrt[23]{P}$ out of the integral.

$p + C_{1} = \sqrt[23]{P} \int \frac{1}{\sqrt[23]{x}} d x$

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 $\sqrt[23]{x}$ as $x^{\frac{1}{23}}$ .

$p + C_{1} = \sqrt[23]{P} \int \frac{1}{x^{\frac{1}{23}}} d x$

Step 2.3.3.2

Move $x^{\frac{1}{23}}$ out of the denominator by raising it to the $- 1$ power.

$p + C_{1} = \sqrt[23]{P} \int {(x^{\frac{1}{23}})}^{- 1} d x$

Step 2.3.3.3

Multiply the exponents in ${(x^{\frac{1}{23}})}^{- 1}$ .

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Step 2.3.3.3.1

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

$p + C_{1} = \sqrt[23]{P} \int x^{\frac{1}{23} \cdot - 1} d x$

Step 2.3.3.3.2

Combine $\frac{1}{23}$ and $- 1$ .

$p + C_{1} = \sqrt[23]{P} \int x^{\frac{- 1}{23}} d x$

Step 2.3.3.3.3

Move the negative in front of the fraction.

$p + C_{1} = \sqrt[23]{P} \int x^{- \frac{1}{23}} d x$

Step 2.3.4

By the Power Rule, the integral of $x^{- \frac{1}{23}}$ with respect to $x$ is $\frac{23}{22} x^{\frac{22}{23}}$ .

$p + C_{1} = \sqrt[23]{P} (\frac{23}{22} x^{\frac{22}{23}} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Rewrite $\sqrt[23]{P} (\frac{23}{22} x^{\frac{22}{23}} + C_{2})$ as $\sqrt[23]{P} \frac{23}{22} x^{\frac{22}{23}} + C_{2}$ .

$p + C_{1} = \sqrt[23]{P} \frac{23}{22} x^{\frac{22}{23}} + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $\sqrt[23]{P}$ and $\frac{23}{22}$ .

$p + C_{1} = \frac{\sqrt[23]{P} \cdot 23}{22} x^{\frac{22}{23}} + C_{2}$

Step 2.3.5.2.2

Move $23$ to the left of $\sqrt[23]{P}$ .

$p + C_{1} = \frac{23 \cdot \sqrt[23]{P}}{22} x^{\frac{22}{23}} + C_{2}$

Step 2.3.5.2.3

Combine $\frac{23 \sqrt[23]{P}}{22}$ and $x^{\frac{22}{23}}$ .

$p + C_{1} = \frac{23 \sqrt[23]{P} x^{\frac{22}{23}}}{22} + C_{2}$

$p + C_{1} = \frac{23}{22} \sqrt[23]{P} x^{\frac{22}{23}} + C_{2}$

Step 2.3.6

Reorder terms.

$p + C_{1} = \frac{23}{22} x^{\frac{22}{23}} \sqrt[23]{P} + C_{2}$

Step 2.4

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

$p = \frac{23}{22} x^{\frac{22}{23}} \sqrt[23]{P} + K$