Calculus Examples

$\frac{d y}{d x} = x^{4} {(x^{5} - 5)}^{2}$ , $y (1) = 5$

Step 1

Rewrite the equation.

$d y = x^{4} {(x^{5} - 5)}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{4} {(x^{5} - 5)}^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x^{4} {(x^{5} - 5)}^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x^{5} - 5$ . Then $d u = 5 x^{4} d x$ , so $\frac{1}{5} d u = x^{4} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = x^{5} - 5$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x^{5} - 5$ .

$\frac{d}{d x} [x^{5} - 5]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $x^{5} - 5$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 5]$

Step 2.3.1.1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$5 x^{4} + 0$

Step 2.3.1.1.5

Add $5 x^{4}$ and $0$ .

$5 x^{4}$

Step 2.3.1.2

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

$y + C_{1} = \int u^{2} \frac{1}{5} d u$

Step 2.3.2

Combine $u^{2}$ and $\frac{1}{5}$ .

$y + C_{1} = \int \frac{u^{2}}{5} d u$

Step 2.3.3

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$y + C_{1} = \frac{1}{5} \int u^{2} d u$

Step 2.3.4

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

$y + C_{1} = \frac{1}{5} (\frac{1}{3} u^{3} + C_{2})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Rewrite $\frac{1}{5} (\frac{1}{3} u^{3} + C_{2})$ as $\frac{1}{5} \cdot \frac{1}{3} u^{3} + C_{2}$ .

$y + C_{1} = \frac{1}{5} \cdot \frac{1}{3} u^{3} + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{3}$ .

$y + C_{1} = \frac{1}{5 \cdot 3} u^{3} + C_{2}$

Step 2.3.5.2.2

Multiply $5$ by $3$ .

$y + C_{1} = \frac{1}{15} u^{3} + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $x^{5} - 5$ .

$y + C_{1} = \frac{1}{15} {(x^{5} - 5)}^{3} + C_{2}$

Step 2.4

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

$y = \frac{1}{15} {(x^{5} - 5)}^{3} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $5$ for $y$ in $y = \frac{1}{15} {(x^{5} - 5)}^{3} + K$ .

$5 = \frac{1}{15} {(1^{5} - 5)}^{3} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{15} \cdot {(1^{5} - 5)}^{3} + K = 5$ .

$\frac{1}{15} \cdot {(1^{5} - 5)}^{3} + K = 5$

Step 4.2

Simplify each term.

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Step 4.2.1

One to any power is one.

$\frac{1}{15} \cdot {(1 - 5)}^{3} + K = 5$

Step 4.2.2

Subtract $5$ from $1$ .

$\frac{1}{15} \cdot {(- 4)}^{3} + K = 5$

Step 4.2.3

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

$\frac{1}{15} \cdot - 64 + K = 5$

Step 4.2.4

Combine $\frac{1}{15}$ and $- 64$ .

$\frac{- 64}{15} + K = 5$

Step 4.2.5

Move the negative in front of the fraction.

$- \frac{64}{15} + K = 5$

Step 4.3

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

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Step 4.3.1

Add $\frac{64}{15}$ to both sides of the equation.

$K = 5 + \frac{64}{15}$

Step 4.3.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$K = 5 \cdot \frac{15}{15} + \frac{64}{15}$

Step 4.3.3

Combine $5$ and $\frac{15}{15}$ .

$K = \frac{5 \cdot 15}{15} + \frac{64}{15}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{5 \cdot 15 + 64}{15}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $5$ by $15$ .

$K = \frac{75 + 64}{15}$

Step 4.3.5.2

Add $75$ and $64$ .

$K = \frac{139}{15}$

Step 5

Substitute $\frac{139}{15}$ for $K$ in $y = \frac{1}{15} {(x^{5} - 5)}^{3} + K$ and simplify.

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Step 5.1

Substitute $\frac{139}{15}$ for $K$ .

$y = \frac{1}{15} {(x^{5} - 5)}^{3} + \frac{139}{15}$

Step 5.2

Simplify each term.

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Step 5.2.1

Use the Binomial Theorem.

$y = \frac{1}{15} ({(x^{5})}^{3} + 3 {(x^{5})}^{2} \cdot - 5 + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply the exponents in ${(x^{5})}^{3}$ .

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Step 5.2.2.1.1

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

$y = \frac{1}{15} (x^{5 \cdot 3} + 3 {(x^{5})}^{2} \cdot - 5 + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.1.2

Multiply $5$ by $3$ .

$y = \frac{1}{15} (x^{15} + 3 {(x^{5})}^{2} \cdot - 5 + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.2

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 5.2.2.2.1

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

$y = \frac{1}{15} (x^{15} + 3 x^{5 \cdot 2} \cdot - 5 + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.2.2

Multiply $5$ by $2$ .

$y = \frac{1}{15} (x^{15} + 3 x^{10} \cdot - 5 + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.3

Multiply $- 5$ by $3$ .

$y = \frac{1}{15} (x^{15} - 15 x^{10} + 3 x^{5} {(- 5)}^{2} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.4

Raise $- 5$ to the power of $2$ .

$y = \frac{1}{15} (x^{15} - 15 x^{10} + 3 x^{5} \cdot 25 + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.5

Multiply $25$ by $3$ .

$y = \frac{1}{15} (x^{15} - 15 x^{10} + 75 x^{5} + {(- 5)}^{3}) + \frac{139}{15}$

Step 5.2.2.6

Raise $- 5$ to the power of $3$ .

$y = \frac{1}{15} (x^{15} - 15 x^{10} + 75 x^{5} - 125) + \frac{139}{15}$

Step 5.2.3

Apply the distributive property.

$y = \frac{1}{15} x^{15} + \frac{1}{15} (- 15 x^{10}) + \frac{1}{15} (75 x^{5}) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Combine $\frac{1}{15}$ and $x^{15}$ .

$y = \frac{x^{15}}{15} + \frac{1}{15} (- 15 x^{10}) + \frac{1}{15} (75 x^{5}) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.2

Cancel the common factor of $15$ .

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Step 5.2.4.2.1

Factor $15$ out of $- 15 x^{10}$ .

$y = \frac{x^{15}}{15} + \frac{1}{15} (15 (- x^{10})) + \frac{1}{15} (75 x^{5}) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.2.2

Cancel the common factor.

$y = \frac{x^{15}}{15} + \frac{1}{15} (15 (- x^{10})) + \frac{1}{15} (75 x^{5}) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.2.3

Rewrite the expression.

$y = \frac{x^{15}}{15} - x^{10} + \frac{1}{15} (75 x^{5}) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.3

Cancel the common factor of $15$ .

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Step 5.2.4.3.1

Factor $15$ out of $75 x^{5}$ .

$y = \frac{x^{15}}{15} - x^{10} + \frac{1}{15} (15 (5 x^{5})) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.3.2

Cancel the common factor.

$y = \frac{x^{15}}{15} - x^{10} + \frac{1}{15} (15 (5 x^{5})) + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.3.3

Rewrite the expression.

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{1}{15} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.4

Cancel the common factor of $5$ .

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Step 5.2.4.4.1

Factor $5$ out of $15$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{1}{5 (3)} \cdot - 125 + \frac{139}{15}$

Step 5.2.4.4.2

Factor $5$ out of $- 125$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{1}{5 \cdot 3} \cdot (5 \cdot - 25) + \frac{139}{15}$

Step 5.2.4.4.3

Cancel the common factor.

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{1}{5 \cdot 3} \cdot (5 \cdot - 25) + \frac{139}{15}$

Step 5.2.4.4.4

Rewrite the expression.

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{1}{3} \cdot - 25 + \frac{139}{15}$

Step 5.2.4.5

Combine $\frac{1}{3}$ and $- 25$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{- 25}{3} + \frac{139}{15}$

Step 5.2.5

Move the negative in front of the fraction.

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} - \frac{25}{3} + \frac{139}{15}$

Step 5.3

To write $- \frac{25}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} - \frac{25}{3} \cdot \frac{5}{5} + \frac{139}{15}$

Step 5.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{25}{3}$ by $\frac{5}{5}$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} - \frac{25 \cdot 5}{3 \cdot 5} + \frac{139}{15}$

Step 5.4.2

Multiply $3$ by $5$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} - \frac{25 \cdot 5}{15} + \frac{139}{15}$

Step 5.5

Combine the numerators over the common denominator.

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{- 25 \cdot 5 + 139}{15}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 25$ by $5$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{- 125 + 139}{15}$

Step 5.6.2

Add $- 125$ and $139$ .

$y = \frac{x^{15}}{15} - x^{10} + 5 x^{5} + \frac{14}{15}$