Calculus Examples

$\frac{d y}{d x} = {(3 x + 2)}^{3}$ , $y (0) = 1$

Step 1

Rewrite the equation.

$d y = {(3 x + 2)}^{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int {(3 x + 2)}^{3} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1.1

Differentiate $3 x + 2$ .

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

Step 2.3.1.1.2

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

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

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

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

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 $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [2]$

Step 2.3.1.1.3.3

Multiply $3$ by $1$ .

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

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

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

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

Step 2.3.2

Combine $u^{3}$ and $\frac{1}{3}$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

Multiply $3$ by $4$ .

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

Step 2.3.6

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

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $1$ for $y$ in $y = \frac{1}{12} {(3 x + 2)}^{4} + K$ .

$1 = \frac{1}{12} {(3 \cdot 0 + 2)}^{4} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{12} \cdot {(3 \cdot 0 + 2)}^{4} + K = 1$ .

$\frac{1}{12} \cdot {(3 \cdot 0 + 2)}^{4} + K = 1$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $3$ by $0$ .

$\frac{1}{12} \cdot {(0 + 2)}^{4} + K = 1$

Step 4.2.2

Add $0$ and $2$ .

$\frac{1}{12} \cdot 2^{4} + K = 1$

Step 4.2.3

Raise $2$ to the power of $4$ .

$\frac{1}{12} \cdot 16 + K = 1$

Step 4.2.4

Cancel the common factor of $4$ .

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Step 4.2.4.1

Factor $4$ out of $12$ .

$\frac{1}{4 (3)} \cdot 16 + K = 1$

Step 4.2.4.2

Factor $4$ out of $16$ .

$\frac{1}{4 \cdot 3} \cdot (4 \cdot 4) + K = 1$

Step 4.2.4.3

Cancel the common factor.

$\frac{1}{4 \cdot 3} \cdot (4 \cdot 4) + K = 1$

Step 4.2.4.4

Rewrite the expression.

$\frac{1}{3} \cdot 4 + K = 1$

Step 4.2.5

Combine $\frac{1}{3}$ and $4$ .

$\frac{4}{3} + K = 1$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $\frac{4}{3}$ from both sides of the equation.

$K = 1 - \frac{4}{3}$

Step 4.3.2

Write $1$ as a fraction with a common denominator.

$K = \frac{3}{3} - \frac{4}{3}$

Step 4.3.3

Combine the numerators over the common denominator.

$K = \frac{3 - 4}{3}$

Step 4.3.4

Subtract $4$ from $3$ .

$K = \frac{- 1}{3}$

Step 4.3.5

Move the negative in front of the fraction.

$K = - \frac{1}{3}$

Step 5

Substitute $- \frac{1}{3}$ for $K$ in $y = \frac{1}{12} {(3 x + 2)}^{4} + K$ and simplify.

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Step 5.1

Substitute $- \frac{1}{3}$ for $K$ .

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

Step 5.2

Simplify each term.

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Step 5.2.1

Use the Binomial Theorem.

$y = \frac{1}{12} ({(3 x)}^{4} + 4 {(3 x)}^{3} \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Apply the product rule to $3 x$ .

$y = \frac{1}{12} (3^{4} x^{4} + 4 {(3 x)}^{3} \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.2

Raise $3$ to the power of $4$ .

$y = \frac{1}{12} (81 x^{4} + 4 {(3 x)}^{3} \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.3

Apply the product rule to $3 x$ .

$y = \frac{1}{12} (81 x^{4} + 4 (3^{3} x^{3}) \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.4

Raise $3$ to the power of $3$ .

$y = \frac{1}{12} (81 x^{4} + 4 (27 x^{3}) \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.5

Multiply $27$ by $4$ .

$y = \frac{1}{12} (81 x^{4} + 108 x^{3} \cdot 2 + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.6

Multiply $2$ by $108$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 6 {(3 x)}^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.7

Apply the product rule to $3 x$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 6 (3^{2} x^{2}) \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.8

Raise $3$ to the power of $2$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 6 (9 x^{2}) \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.9

Multiply $9$ by $6$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 54 x^{2} \cdot 2^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.10

Raise $2$ to the power of $2$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 54 x^{2} \cdot 4 + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.11

Multiply $4$ by $54$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 216 x^{2} + 4 (3 x) \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.12

Multiply $3$ by $4$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 216 x^{2} + 12 x \cdot 2^{3} + 2^{4}) - \frac{1}{3}$

Step 5.2.2.13

Raise $2$ to the power of $3$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 216 x^{2} + 12 x \cdot 8 + 2^{4}) - \frac{1}{3}$

Step 5.2.2.14

Multiply $8$ by $12$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 216 x^{2} + 96 x + 2^{4}) - \frac{1}{3}$

Step 5.2.2.15

Raise $2$ to the power of $4$ .

$y = \frac{1}{12} (81 x^{4} + 216 x^{3} + 216 x^{2} + 96 x + 16) - \frac{1}{3}$

Step 5.2.3

Apply the distributive property.

$y = \frac{1}{12} (81 x^{4}) + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Cancel the common factor of $3$ .

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Step 5.2.4.1.1

Factor $3$ out of $12$ .

$y = \frac{1}{3 (4)} (81 x^{4}) + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.1.2

Factor $3$ out of $81 x^{4}$ .

$y = \frac{1}{3 (4)} (3 (27 x^{4})) + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.1.3

Cancel the common factor.

$y = \frac{1}{3 \cdot 4} (3 (27 x^{4})) + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.1.4

Rewrite the expression.

$y = \frac{1}{4} (27 x^{4}) + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.2

Combine $27$ and $\frac{1}{4}$ .

$y = \frac{27}{4} x^{4} + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.3

Combine $\frac{27}{4}$ and $x^{4}$ .

$y = \frac{27 x^{4}}{4} + \frac{1}{12} (216 x^{3}) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.4

Cancel the common factor of $12$ .

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Step 5.2.4.4.1

Factor $12$ out of $216 x^{3}$ .

$y = \frac{27 x^{4}}{4} + \frac{1}{12} (12 (18 x^{3})) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.4.2

Cancel the common factor.

$y = \frac{27 x^{4}}{4} + \frac{1}{12} (12 (18 x^{3})) + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.4.3

Rewrite the expression.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + \frac{1}{12} (216 x^{2}) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.5

Cancel the common factor of $12$ .

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Step 5.2.4.5.1

Factor $12$ out of $216 x^{2}$ .

$y = \frac{27 x^{4}}{4} + 18 x^{3} + \frac{1}{12} (12 (18 x^{2})) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.5.2

Cancel the common factor.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + \frac{1}{12} (12 (18 x^{2})) + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.5.3

Rewrite the expression.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + \frac{1}{12} (96 x) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.6

Cancel the common factor of $12$ .

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Step 5.2.4.6.1

Factor $12$ out of $96 x$ .

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + \frac{1}{12} (12 (8 x)) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.6.2

Cancel the common factor.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + \frac{1}{12} (12 (8 x)) + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.6.3

Rewrite the expression.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{1}{12} \cdot 16 - \frac{1}{3}$

Step 5.2.4.7

Cancel the common factor of $4$ .

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Step 5.2.4.7.1

Factor $4$ out of $12$ .

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{1}{4 (3)} \cdot 16 - \frac{1}{3}$

Step 5.2.4.7.2

Factor $4$ out of $16$ .

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{1}{4 \cdot 3} \cdot (4 \cdot 4) - \frac{1}{3}$

Step 5.2.4.7.3

Cancel the common factor.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{1}{4 \cdot 3} \cdot (4 \cdot 4) - \frac{1}{3}$

Step 5.2.4.7.4

Rewrite the expression.

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{1}{3} \cdot 4 - \frac{1}{3}$

Step 5.2.4.8

Combine $\frac{1}{3}$ and $4$ .

$y = \frac{27 x^{4}}{4} + 18 x^{3} + 18 x^{2} + 8 x + \frac{4}{3} - \frac{1}{3}$

Step 5.3

Combine the numerators over the common denominator.

$y = 18 x^{3} + 18 x^{2} + 8 x + \frac{27 x^{4}}{4} + \frac{4 - 1}{3}$

Step 5.4

Subtract $1$ from $4$ .

$y = 18 x^{3} + 18 x^{2} + 8 x + \frac{27 x^{4}}{4} + \frac{3}{3}$

Step 5.5

Divide $3$ by $3$ .

$y = 18 x^{3} + 18 x^{2} + 8 x + \frac{27 x^{4}}{4} + 1$