Calculus Examples

$\frac{d y}{d x} = 4 x {(x^{2} + 8)}^{- \frac{1}{3}}$ , $y (0) = 0$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 x {(x^{2} + 8)}^{- \frac{1}{3}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 x {(x^{2} + 8)}^{- \frac{1}{3}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = 4 \int x {(x^{2} + 8)}^{- \frac{1}{3}} d x$

Step 2.3.2

Let $u = x^{2} + 8$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x^{2} + 8$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{2} + 8$ .

$\frac{d}{d x} [x^{2} + 8]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8]$

Step 2.3.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [8]$

Step 2.3.2.1.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.2.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Move $u^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.4

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

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

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 $\frac{1}{2}$ and $4$ .

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

Step 2.3.5.1.2

Cancel the common factor of $4$ and $2$ .

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Step 2.3.5.1.2.1

Factor $2$ out of $4$ .

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

Step 2.3.5.1.2.2

Cancel the common factors.

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Step 2.3.5.1.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.5.1.2.2.2

Cancel the common factor.

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

Step 2.3.5.1.2.2.3

Rewrite the expression.

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

Step 2.3.5.1.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.5.2

Apply basic rules of exponents.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

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

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

Step 2.3.5.2.2.2

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

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

Step 2.3.5.2.2.3

Move the negative in front of the fraction.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Multiply $2$ by $3$ .

$y + C_{1} = \frac{6}{2} u^{\frac{2}{3}} + C_{2}$

Step 2.3.7.2.3

Cancel the common factor of $6$ and $2$ .

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Step 2.3.7.2.3.1

Factor $2$ out of $6$ .

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

Step 2.3.7.2.3.2

Cancel the common factors.

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Step 2.3.7.2.3.2.1

Factor $2$ out of $2$ .

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

Step 2.3.7.2.3.2.2

Cancel the common factor.

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

Step 2.3.7.2.3.2.3

Rewrite the expression.

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

Step 2.3.7.2.3.2.4

Divide $3$ by $1$ .

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

Step 2.3.8

Replace all occurrences of $u$ with $x^{2} + 8$ .

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $0$ for $y$ in $y = 3 {(x^{2} + 8)}^{\frac{2}{3}} + K$ .

$0 = 3 {(0^{2} + 8)}^{\frac{2}{3}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $3 {(0^{2} + 8)}^{\frac{2}{3}} + K = 0$ .

$3 {(0^{2} + 8)}^{\frac{2}{3}} + K = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$3 {(0 + 8)}^{\frac{2}{3}} + K = 0$

Step 4.2.2

Add $0$ and $8$ .

$3 \cdot 8^{\frac{2}{3}} + K = 0$

Step 4.2.3

Rewrite $8$ as $2^{3}$ .

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

Step 4.2.4

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

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

Step 4.2.5

Cancel the common factor of $3$ .

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Rewrite the expression.

$3 \cdot 2^{2} + K = 0$

Step 4.2.6

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

$3 \cdot 4 + K = 0$

Step 4.2.7

Multiply $3$ by $4$ .

$12 + K = 0$

Step 4.3

Subtract $12$ from both sides of the equation.

$K = - 12$

Step 5

Substitute $- 12$ for $K$ in $y = 3 {(x^{2} + 8)}^{\frac{2}{3}} + K$ and simplify.

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Step 5.1

Substitute $- 12$ for $K$ .

$y = 3 {(x^{2} + 8)}^{\frac{2}{3}} - 12$