Calculus Examples

$\frac{d y}{d t} = \frac{t^{2} \sqrt{y}}{1 + t^{3}}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d t} = \frac{t^{2}}{1 + t^{3}} \sqrt{y}$

Step 1.2

Multiply both sides by $\frac{1}{\sqrt{y}}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} (\frac{t^{2}}{1 + t^{3}} \sqrt{y})$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $\sqrt{y}$ .

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Step 1.3.1.1

Factor $\sqrt{y}$ out of $\frac{t^{2}}{1 + t^{3}} \sqrt{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} (\sqrt{y} \frac{t^{2}}{1 + t^{3}})$

Step 1.3.1.2

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} (\sqrt{y} \frac{t^{2}}{1 + t^{3}})$

Step 1.3.1.3

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{t^{2}}{1 + t^{3}}$

Step 1.3.2

Simplify the denominator.

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Step 1.3.2.1

Rewrite $1$ as $1^{3}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{t^{2}}{1^{3} + t^{3}}$

Step 1.3.2.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = t$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{t^{2}}{(1 + t) (1^{2} - 1 t + t^{2})}$

Step 1.3.2.3

Simplify.

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Step 1.3.2.3.1

One to any power is one.

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{t^{2}}{(1 + t) (1 - 1 t + t^{2})}$

Step 1.3.2.3.2

Rewrite $- 1 t$ as $- t$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{t^{2}}{(1 + t) (1 - t + t^{2})}$

Step 1.4

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = \frac{t^{2}}{(1 + t) (1 - t + t^{2})} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y}} d y = \int \frac{t^{2}}{(1 + t) (1 - t + t^{2})} d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int \frac{t^{2}}{(1 + t) (1 - t + t^{2})} d t$

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = (1 + t) (1 - t + t^{2})$ . Then $d u = 3 t^{2} d t$ , so $\frac{1}{3} d u = t^{2} d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = (1 + t) (1 - t + t^{2})$ . Find $\frac{d u}{d t}$ .

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Step 2.3.1.1.1

Differentiate $(1 + t) (1 - t + t^{2})$ .

$\frac{d}{d t} [(1 + t) (1 - t + t^{2})]$

Step 2.3.1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 1 + t$ and $g (t) = 1 - t + t^{2}$ .

$(1 + t) \frac{d}{d t} [1 - t + t^{2}] + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3

Differentiate.

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Step 2.3.1.1.3.1

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

$(1 + t) (\frac{d}{d t} [1] + \frac{d}{d t} [- t] + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.2

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

$(1 + t) (0 + \frac{d}{d t} [- t] + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.3

Add $0$ and $\frac{d}{d t} [- t]$ .

$(1 + t) (\frac{d}{d t} [- t] + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

$(1 + t) (- \frac{d}{d t} [t] + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.5

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

$(1 + t) (- 1 \cdot 1 + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.6

Multiply $- 1$ by $1$ .

$(1 + t) (- 1 + \frac{d}{d t} [t^{2}]) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.7

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

$(1 + t) (- 1 + 2 t) + (1 - t + t^{2}) \frac{d}{d t} [1 + t]$

Step 2.3.1.1.3.8

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

$(1 + t) (- 1 + 2 t) + (1 - t + t^{2}) (\frac{d}{d t} [1] + \frac{d}{d t} [t])$

Step 2.3.1.1.3.9

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

$(1 + t) (- 1 + 2 t) + (1 - t + t^{2}) (0 + \frac{d}{d t} [t])$

Step 2.3.1.1.3.10

Add $0$ and $\frac{d}{d t} [t]$ .

$(1 + t) (- 1 + 2 t) + (1 - t + t^{2}) \frac{d}{d t} [t]$

Step 2.3.1.1.3.11

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

$(1 + t) (- 1 + 2 t) + (1 - t + t^{2}) \cdot 1$

Step 2.3.1.1.3.12

Multiply $1 - t + t^{2}$ by $1$ .

$(1 + t) (- 1 + 2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4

Simplify.

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Step 2.3.1.1.4.1

Apply the distributive property.

$(1 + t) \cdot - 1 + (1 + t) (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.2

Apply the distributive property.

$1 \cdot - 1 + t \cdot - 1 + (1 + t) (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.3

Apply the distributive property.

$1 \cdot - 1 + t \cdot - 1 + 1 (2 t) + t (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4

Combine terms.

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Step 2.3.1.1.4.4.1

Multiply $- 1$ by $1$ .

$- 1 + t \cdot - 1 + 1 (2 t) + t (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.2

Move $- 1$ to the left of $t$ .

$- 1 - 1 \cdot t + 1 (2 t) + t (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.3

Rewrite $- 1 t$ as $- t$ .

$- 1 - t + 1 (2 t) + t (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.4

Multiply $2$ by $1$ .

$- 1 - t + 2 t + t (2 t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.5

Raise $t$ to the power of $1$ .

$- 1 - t + 2 t + 2 (t^{1} t) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.6

Raise $t$ to the power of $1$ .

$- 1 - t + 2 t + 2 (t^{1} t^{1}) + 1 - t + t^{2}$

Step 2.3.1.1.4.4.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 - t + 2 t + 2 t^{1 + 1} + 1 - t + t^{2}$

Step 2.3.1.1.4.4.8

Add $1$ and $1$ .

$- 1 - t + 2 t + 2 t^{2} + 1 - t + t^{2}$

Step 2.3.1.1.4.4.9

Add $- t$ and $2 t$ .

$- 1 + t + 2 t^{2} + 1 - t + t^{2}$

Step 2.3.1.1.4.4.10

Add $- 1$ and $1$ .

$t + 2 t^{2} + 0 - t + t^{2}$

Step 2.3.1.1.4.4.11

Add $t + 2 t^{2}$ and $0$ .

$t + 2 t^{2} - t + t^{2}$

Step 2.3.1.1.4.4.12

Subtract $t$ from $t$ .

$2 t^{2} + 0 + t^{2}$

Step 2.3.1.1.4.4.13

Add $2 t^{2}$ and $0$ .

$2 t^{2} + t^{2}$

Step 2.3.1.1.4.4.14

Add $2 t^{2}$ and $t^{2}$ .

$3 t^{2}$

Step 2.3.1.2

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

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

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

Step 2.3.2.2

Move $3$ to the left of $u$ .

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

Step 2.3.3

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

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

Step 2.3.4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

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

Step 2.3.5

Simplify.

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

Step 2.3.6

Replace all occurrences of $u$ with $(1 + t) (1 - t + t^{2})$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |) + C_{2}$

Step 2.4

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

$2 y^{\frac{1}{2}} = \frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |) + C$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |) + C$ by $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |)}{2} + \frac{C}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |)}{2} + \frac{C}{2}$

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |)}{2} + \frac{C}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Combine the numerators over the common denominator.

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| (1 + t) (1 - t + t^{2}) |) + C}{2}$

Step 3.1.3.2

Simplify each term.

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Step 3.1.3.2.1

Expand $(1 + t) (1 - t + t^{2})$ by multiplying each term in the first expression by each term in the second expression.

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 \cdot 1 + 1 (- t) + 1 t^{2} + t \cdot 1 + t (- t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2

Simplify each term.

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Step 3.1.3.2.2.1

Multiply $1$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 + 1 (- t) + 1 t^{2} + t \cdot 1 + t (- t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.2

Multiply $- t$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + 1 t^{2} + t \cdot 1 + t (- t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.3

Multiply $t^{2}$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t \cdot 1 + t (- t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.4

Multiply $t$ by $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t + t (- t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.5

Rewrite using the commutative property of multiplication.

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - t \cdot t + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.6

Multiply $t$ by $t$ by adding the exponents.

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Step 3.1.3.2.2.6.1

Move $t$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - (t \cdot t) + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.6.2

Multiply $t$ by $t$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - t^{2} + t \cdot t^{2} |) + C}{2}$

Step 3.1.3.2.2.7

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 3.1.3.2.2.7.1

Multiply $t$ by $t^{2}$ .

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Step 3.1.3.2.2.7.1.1

Raise $t$ to the power of $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - t^{2} + t^{1} t^{2} |) + C}{2}$

Step 3.1.3.2.2.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - t^{2} + t^{1 + 2} |) + C}{2}$

Step 3.1.3.2.2.7.2

Add $1$ and $2$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 - t + t^{2} + t - t^{2} + t^{3} |) + C}{2}$

Step 3.1.3.2.3

Combine the opposite terms in $1 - t + t^{2} + t - t^{2} + t^{3}$ .

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Step 3.1.3.2.3.1

Add $- t$ and $t$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 + t^{2} + 0 - t^{2} + t^{3} |) + C}{2}$

Step 3.1.3.2.3.2

Add $1 + t^{2}$ and $0$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 + t^{2} - t^{2} + t^{3} |) + C}{2}$

Step 3.1.3.2.3.3

Subtract $t^{2}$ from $t^{2}$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 + 0 + t^{3} |) + C}{2}$

Step 3.1.3.2.3.4

Add $1$ and $0$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} \ln (| 1 + t^{3} |) + C}{2}$

Step 3.1.3.2.4

Simplify $\frac{1}{3} \ln (| 1 + t^{3} |)$ by moving $\frac{1}{3}$ inside the logarithm.

$y^{\frac{1}{2}} = \frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$

Step 3.3.1.1.2

Simplify.

$y = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2})}^{2}$ .

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Step 3.3.2.1.1

Split the fraction $\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + C}{2}$ into two fractions.

$y = {(\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}})}{2} + \frac{C}{2})}^{2}$

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

Rewrite $\frac{\ln ({| 1 + t^{3} |}^{\frac{1}{3}})}{2}$ as $\frac{1}{2} \ln ({| 1 + t^{3} |}^{\frac{1}{3}})$ .

$y = {(\frac{1}{2} \ln ({| 1 + t^{3} |}^{\frac{1}{3}}) + \frac{C}{2})}^{2}$

Step 3.3.2.1.2.2

Simplify $\frac{1}{2} \ln ({| 1 + t^{3} |}^{\frac{1}{3}})$ by moving $\frac{1}{2}$ inside the logarithm.

$y = {(\ln ({({| 1 + t^{3} |}^{\frac{1}{3}})}^{\frac{1}{2}}) + \frac{C}{2})}^{2}$

Step 3.3.2.1.2.3

Multiply the exponents in ${({| 1 + t^{3} |}^{\frac{1}{3}})}^{\frac{1}{2}}$ .

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Step 3.3.2.1.2.3.1

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

$y = {(\ln ({| 1 + t^{3} |}^{\frac{1}{3} \cdot \frac{1}{2}}) + \frac{C}{2})}^{2}$

Step 3.3.2.1.2.3.2

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

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Step 3.3.2.1.2.3.2.1

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

$y = {(\ln ({| 1 + t^{3} |}^{\frac{1}{3 \cdot 2}}) + \frac{C}{2})}^{2}$

Step 3.3.2.1.2.3.2.2

Multiply $3$ by $2$ .

$y = {(\ln ({| 1 + t^{3} |}^{\frac{1}{6}}) + \frac{C}{2})}^{2}$

Step 4

Simplify the constant of integration.

$y = {(\ln ({| 1 + t^{3} |}^{\frac{1}{6}}) + C)}^{2}$