Calculus Examples

$\frac{d y}{d x} = \sqrt[3]{3 x - 1}$

Step 1

Rewrite the equation.

$d y = \sqrt[3]{3 x - 1} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sqrt[3]{3 x - 1} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sqrt[3]{3 x - 1} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 3 x - 1$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 3 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $3 x - 1$ .

$\frac{d}{d x} [3 x - 1]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$

Step 2.3.1.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.3.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 2.3.1.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [- 1]$

Step 2.3.1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 1]$

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

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

$3 + 0$

Step 2.3.1.1.4.2

Add $3$ and $0$ .

$3$

Step 2.3.1.2

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

$y + C_{1} = \int \sqrt[3]{u} \frac{1}{3} d u$

Step 2.3.2

Combine $\sqrt[3]{u}$ and $\frac{1}{3}$ .

$y + C_{1} = \int \frac{\sqrt[3]{u}}{3} d u$

Step 2.3.3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$y + C_{1} = \frac{1}{3} \int \sqrt[3]{u} d u$

Step 2.3.4

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

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

Step 2.3.5

By the Power Rule, the integral of $u^{\frac{1}{3}}$ with respect to $u$ is $\frac{3}{4} u^{\frac{4}{3}}$ .

$y + C_{1} = \frac{1}{3} (\frac{3}{4} u^{\frac{4}{3}} + C_{2})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Rewrite $\frac{1}{3} (\frac{3}{4} u^{\frac{4}{3}} + C_{2})$ as $\frac{1}{3} \cdot \frac{3}{4} u^{\frac{4}{3}} + C_{2}$ .

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

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{3}$ by $\frac{3}{4}$ .

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

Step 2.3.6.2.2

Multiply $3$ by $4$ .

$y + C_{1} = \frac{3}{12} u^{\frac{4}{3}} + C_{2}$

Step 2.3.6.2.3

Cancel the common factor of $3$ and $12$ .

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Step 2.3.6.2.3.1

Factor $3$ out of $3$ .

$y + C_{1} = \frac{3 (1)}{12} u^{\frac{4}{3}} + C_{2}$

Step 2.3.6.2.3.2

Cancel the common factors.

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Step 2.3.6.2.3.2.1

Factor $3$ out of $12$ .

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

Step 2.3.6.2.3.2.2

Cancel the common factor.

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

Step 2.3.6.2.3.2.3

Rewrite the expression.

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

Step 2.3.7

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 2.4

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

$y = \frac{1}{4} {(3 x - 1)}^{\frac{4}{3}} + K$