Calculus Examples

$\frac{d y}{d x} = \frac{- 24}{{(2 x + 1)}^{3}}$

Step 1

Rewrite the equation.

$d y = \frac{- 24}{{(2 x + 1)}^{3}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{- 24}{{(2 x + 1)}^{3}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{- 24}{{(2 x + 1)}^{3}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y + C_{1} = \int - \frac{24}{{(2 x + 1)}^{3}} d x$

Step 2.3.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - \int \frac{24}{{(2 x + 1)}^{3}} d x$

Step 2.3.3

Since $24$ is constant with respect to $x$ , move $24$ out of the integral.

$y + C_{1} = - (24 \int \frac{1}{{(2 x + 1)}^{3}} d x)$

Step 2.3.4

Multiply $24$ by $- 1$ .

$y + C_{1} = - 24 \int \frac{1}{{(2 x + 1)}^{3}} d x$

Step 2.3.5

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

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Step 2.3.5.1

Let $u = 2 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.5.1.1

Differentiate $2 x + 1$ .

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

Step 2.3.5.1.2

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 2.3.5.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.3.5.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

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

Step 2.3.5.1.3.3

Multiply $2$ by $1$ .

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

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 $0$ .

$2 + 0$

Step 2.3.5.1.4.2

Add $2$ and $0$ .

$2$

Step 2.3.5.2

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

$y + C_{1} = - 24 \int \frac{1}{u^{3}} \cdot \frac{1}{2} d u$

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$y + C_{1} = - 24 \int \frac{1}{u^{3} \cdot 2} d u$

Step 2.3.6.2

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

$y + C_{1} = - 24 \int \frac{1}{2 u^{3}} d u$

Step 2.3.7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = - 24 (\frac{1}{2} \int \frac{1}{u^{3}} d u)$

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 $\frac{1}{2}$ and $- 24$ .

$y + C_{1} = \frac{- 24}{2} \int \frac{1}{u^{3}} d u$

Step 2.3.8.1.2

Cancel the common factor of $- 24$ and $2$ .

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Step 2.3.8.1.2.1

Factor $2$ out of $- 24$ .

$y + C_{1} = \frac{2 \cdot - 12}{2} \int \frac{1}{u^{3}} d u$

Step 2.3.8.1.2.2

Cancel the common factors.

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Step 2.3.8.1.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot - 12}{2 (1)} \int \frac{1}{u^{3}} d u$

Step 2.3.8.1.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot - 12}{2 \cdot 1} \int \frac{1}{u^{3}} d u$

Step 2.3.8.1.2.2.3

Rewrite the expression.

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

Step 2.3.8.1.2.2.4

Divide $- 12$ by $1$ .

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

Step 2.3.8.2

Apply basic rules of exponents.

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Step 2.3.8.2.1

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

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

Step 2.3.8.2.2

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

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Step 2.3.8.2.2.1

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

$y + C_{1} = - 12 \int u^{3 \cdot - 1} d u$

Step 2.3.8.2.2.2

Multiply $3$ by $- 1$ .

$y + C_{1} = - 12 \int u^{- 3} d u$

Step 2.3.9

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

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

Step 2.3.10

Simplify.

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Step 2.3.10.1

Simplify.

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Step 2.3.10.1.1

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

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

Step 2.3.10.1.2

Move $u^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.10.2

Simplify.

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

Step 2.3.10.3

Simplify.

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Step 2.3.10.3.1

Multiply $- 1$ by $- 12$ .

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

Step 2.3.10.3.2

Combine $12$ and $\frac{1}{2 u^{2}}$ .

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

Step 2.3.10.3.3

Cancel the common factor of $12$ and $2$ .

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Step 2.3.10.3.3.1

Factor $2$ out of $12$ .

$y + C_{1} = \frac{2 \cdot 6}{2 u^{2}} + C_{2}$

Step 2.3.10.3.3.2

Cancel the common factors.

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Step 2.3.10.3.3.2.1

Factor $2$ out of $2 u^{2}$ .

$y + C_{1} = \frac{2 \cdot 6}{2 (u^{2})} + C_{2}$

Step 2.3.10.3.3.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 6}{2 u^{2}} + C_{2}$

Step 2.3.10.3.3.2.3

Rewrite the expression.

$y + C_{1} = \frac{6}{u^{2}} + C_{2}$

Step 2.3.11

Replace all occurrences of $u$ with $2 x + 1$ .

$y + C_{1} = \frac{6}{{(2 x + 1)}^{2}} + C_{2}$

Step 2.4

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

$y = \frac{6}{{(2 x + 1)}^{2}} + K$