Calculus Examples

$\frac{d y}{d x} = \frac{12 x^{2}}{\sqrt{2 + x^{3}}}$

Step 1

Rewrite the equation.

$d y = \frac{12 x^{2}}{\sqrt{2 + x^{3}}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{12 x^{2}}{\sqrt{2 + x^{3}}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{12 x^{2}}{\sqrt{2 + x^{3}}} d x$

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.

$y + C_{1} = 12 \int \frac{x^{2}}{\sqrt{2 + x^{3}}} d x$

Step 2.3.2

Let $u = 2 + x^{3}$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 2 + x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $2 + x^{3}$ .

$\frac{d}{d x} [2 + x^{3}]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $2 + x^{3}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [x^{3}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [x^{3}]$

Step 2.3.2.1.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{3}]$

Step 2.3.2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 + 3 x^{2}$

Step 2.3.2.1.5

Add $0$ and $3 x^{2}$ .

$3 x^{2}$

Step 2.3.2.2

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

$y + C_{1} = 12 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{3} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

$y + C_{1} = 12 \int \frac{1}{\sqrt{u} \cdot 3} d u$

Step 2.3.3.2

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

$y + C_{1} = 12 \int \frac{1}{3 \sqrt{u}} d u$

Step 2.3.4

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

$y + C_{1} = 12 (\frac{1}{3} \int \frac{1}{\sqrt{u}} d u)$

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 $\frac{1}{3}$ and $12$ .

$y + C_{1} = \frac{12}{3} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2

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

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Step 2.3.5.1.2.1

Factor $3$ out of $12$ .

$y + C_{1} = \frac{3 \cdot 4}{3} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2

Cancel the common factors.

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Step 2.3.5.1.2.2.1

Factor $3$ out of $3$ .

$y + C_{1} = \frac{3 \cdot 4}{3 (1)} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{3 \cdot 4}{3 \cdot 1} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{4}{1} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2.4

Divide $4$ by $1$ .

$y + C_{1} = 4 \int \frac{1}{\sqrt{u}} d u$

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 $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.3.5.2.2

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

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

Step 2.3.5.2.3

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

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Step 2.3.5.2.3.1

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

$y + C_{1} = 4 \int u^{\frac{1}{2} \cdot - 1} d u$

Step 2.3.5.2.3.2

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

$y + C_{1} = 4 \int u^{\frac{- 1}{2}} d u$

Step 2.3.5.2.3.3

Move the negative in front of the fraction.

$y + C_{1} = 4 \int u^{- \frac{1}{2}} d u$

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Multiply $4$ by $2$ .

$y + C_{1} = 8 u^{\frac{1}{2}} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $2 + x^{3}$ .

$y + C_{1} = 8 {(2 + x^{3})}^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$y = 8 {(2 + x^{3})}^{\frac{1}{2}} + K$