Calculus Examples

$\frac{d y}{d x} = 4 x + \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}}$

Step 1

Rewrite the equation.

$d y = (4 x + \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 x + \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 x + \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 4 x d x + \int \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int x d x + \int \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.3.3

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

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + \int \frac{9 x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.3.4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + 9 \int \frac{x^{2}}{{(3 x^{3} + 1)}^{\frac{3}{2}}} d x$

Step 2.3.5

Let $u = 3 x^{3} + 1$ . Then $d u = 9 x^{2} d x$ , so $\frac{1}{9} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.5.1

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

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Step 2.3.5.1.1

Differentiate $3 x^{3} + 1$ .

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

Step 2.3.5.1.2

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

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

Step 2.3.5.1.3

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

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Step 2.3.5.1.3.1

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

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

Step 2.3.5.1.3.2

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

$3 (3 x^{2}) + \frac{d}{d x} [1]$

Step 2.3.5.1.3.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [1]$

Step 2.3.5.1.4

Differentiate using the Constant Rule.

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Step 2.3.5.1.4.1

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

$9 x^{2} + 0$

Step 2.3.5.1.4.2

Add $9 x^{2}$ and $0$ .

$9 x^{2}$

Step 2.3.5.2

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

$y + C_{1} = 4 (\frac{1}{2} x^{2} + C_{2}) + 9 \int \frac{1}{u^{\frac{3}{2}}} \cdot \frac{1}{9} d u$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + 9 \int \frac{1}{u^{\frac{3}{2}}} \cdot \frac{1}{9} d u$

Step 2.3.6.2

Multiply $\frac{1}{u^{\frac{3}{2}}}$ by $\frac{1}{9}$ .

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + 9 \int \frac{1}{u^{\frac{3}{2}} \cdot 9} d u$

Step 2.3.6.3

Move $9$ to the left of $u^{\frac{3}{2}}$ .

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + 9 \int \frac{1}{9 u^{\frac{3}{2}}} d u$

Step 2.3.7

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + 9 (\frac{1}{9} \int \frac{1}{u^{\frac{3}{2}}} d u)$

Step 2.3.8

Simplify the expression.

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Step 2.3.8.1

Simplify.

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Step 2.3.8.1.1

Combine $\frac{1}{9}$ and $9$ .

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + \frac{9}{9} \int \frac{1}{u^{\frac{3}{2}}} d u$

Step 2.3.8.1.2

Cancel the common factor of $9$ .

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Step 2.3.8.1.2.1

Cancel the common factor.

$y + C_{1} = 4 (\frac{x^{2}}{2} + C_{2}) + \frac{9}{9} \int \frac{1}{u^{\frac{3}{2}}} d u$

Step 2.3.8.1.2.2

Rewrite the expression.

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

Step 2.3.8.1.3

Multiply $\int \frac{1}{u^{\frac{3}{2}}} d u$ by $1$ .

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

Step 2.3.8.2

Apply basic rules of exponents.

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Step 2.3.8.2.1

Move $u^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3.8.2.2

Multiply the exponents in ${(u^{\frac{3}{2}})}^{- 1}$ .

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Step 2.3.8.2.2.1

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

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

Step 2.3.8.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 2.3.8.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 2.3.8.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.3.8.2.2.3

Move the negative in front of the fraction.

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

Step 2.3.9

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

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

Step 2.3.10

Simplify.

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

Step 2.3.11

Replace all occurrences of $u$ with $3 x^{3} + 1$ .

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

Step 2.4

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

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