Calculus Examples

$\frac{d C}{d x} = \frac{12}{\sqrt[3]{12 x + 1}}$

Step 1

Rewrite the equation.

$d C = \frac{12}{\sqrt[3]{12 x + 1}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d C = \int \frac{12}{\sqrt[3]{12 x + 1}} d x$

Step 2.2

Rewrite.

$C = \int \frac{12}{\sqrt[3]{12 x + 1}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$C = 12 \int \frac{1}{\sqrt[3]{12 x + 1}} d x$

Step 2.3.2

Let $u = 12 x + 1$ . Then $d u = 12 d x$ , so $\frac{1}{12} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 12 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $12 x + 1$ .

$\frac{d}{d x} [12 x + 1]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $12 x + 1$ with respect to $x$ is $\frac{d}{d x} [12 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 2.3.2.1.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$12 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.3.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$12 \cdot 1 + \frac{d}{d x} [1]$

Step 2.3.2.1.3.3

Multiply $12$ by $1$ .

$12 + \frac{d}{d x} [1]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$12 + 0$

Step 2.3.2.1.4.2

Add $12$ and $0$ .

$12$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$C = 12 \int \frac{1}{\sqrt[3]{u}} \cdot \frac{1}{12} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{\sqrt[3]{u}}$ by $\frac{1}{12}$ .

$C = 12 \int \frac{1}{\sqrt[3]{u} \cdot 12} d u$

Step 2.3.3.2

Move $12$ to the left of $\sqrt[3]{u}$ .

$C = 12 \int \frac{1}{12 \sqrt[3]{u}} d u$

Step 2.3.4

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$C = 12 (\frac{1}{12} \int \frac{1}{\sqrt[3]{u}} d u)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Simplify.

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Step 2.3.5.1.1

Combine $\frac{1}{12}$ and $12$ .

$C = \frac{12}{12} \int \frac{1}{\sqrt[3]{u}} d u$

Step 2.3.5.1.2

Cancel the common factor of $12$ .

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Step 2.3.5.1.2.1

Cancel the common factor.

$C = \frac{12}{12} \int \frac{1}{\sqrt[3]{u}} d u$

Step 2.3.5.1.2.2

Rewrite the expression.

$C = 1 \int \frac{1}{\sqrt[3]{u}} d u$

Step 2.3.5.1.3

Multiply $\int \frac{1}{\sqrt[3]{u}} d u$ by $1$ .

$C = \int \frac{1}{\sqrt[3]{u}} d u$

Step 2.3.5.2

Apply basic rules of exponents.

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Step 2.3.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u}$ as $u^{\frac{1}{3}}$ .

$C = \int \frac{1}{u^{\frac{1}{3}}} d u$

Step 2.3.5.2.2

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$C = \int {(u^{\frac{1}{3}})}^{- 1} d u$

Step 2.3.5.2.3

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 2.3.5.2.3.1

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

$C = \int u^{\frac{1}{3} \cdot - 1} d u$

Step 2.3.5.2.3.2

Combine $\frac{1}{3}$ and $- 1$ .

$C = \int u^{\frac{- 1}{3}} d u$

Step 2.3.5.2.3.3

Move the negative in front of the fraction.

$C = \int u^{- \frac{1}{3}} d u$

Step 2.3.6

By the Power Rule, the integral of $u^{- \frac{1}{3}}$ with respect to $u$ is $\frac{3}{2} u^{\frac{2}{3}}$ .

$C = \frac{3}{2} u^{\frac{2}{3}} + C_{1}$

Step 2.3.7

Replace all occurrences of $u$ with $12 x + 1$ .

$C = \frac{3}{2} {(12 x + 1)}^{\frac{2}{3}} + C_{1}$

Step 2.4

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

$C = \frac{3}{2} {(12 x + 1)}^{\frac{2}{3}} + K$