Calculus Examples

$\frac{d s}{d t} = 8 t {(3 t^{2} - 5)}^{3}$

Step 1

Rewrite the equation.

$d s = 8 t {(3 t^{2} - 5)}^{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 8 t {(3 t^{2} - 5)}^{3} d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 8 t {(3 t^{2} - 5)}^{3} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $8$ is constant with respect to $t$ , move $8$ out of the integral.

$s + C_{1} = 8 \int t {(3 t^{2} - 5)}^{3} d t$

Step 2.3.2

Let $u = 3 t^{2} - 5$ . Then $d u = 6 t d t$ , so $\frac{1}{6} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 3 t^{2} - 5$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $3 t^{2} - 5$ .

$\frac{d}{d t} [3 t^{2} - 5]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $3 t^{2} - 5$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [- 5]$ .

$\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [- 5]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d t} [3 t^{2}]$ .

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Step 2.3.2.1.3.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{2}$ with respect to $t$ is $3 \frac{d}{d t} [t^{2}]$ .

$3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 5]$

Step 2.3.2.1.3.2

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

$3 (2 t) + \frac{d}{d t} [- 5]$

Step 2.3.2.1.3.3

Multiply $2$ by $3$ .

$6 t + \frac{d}{d t} [- 5]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

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

$6 t + 0$

Step 2.3.2.1.4.2

Add $6 t$ and $0$ .

$6 t$

Step 2.3.2.2

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

$s + C_{1} = 8 \int u^{3} \frac{1}{6} d u$

Step 2.3.3

Combine $u^{3}$ and $\frac{1}{6}$ .

$s + C_{1} = 8 \int \frac{u^{3}}{6} d u$

Step 2.3.4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$s + C_{1} = 8 (\frac{1}{6} \int u^{3} d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{6}$ and $8$ .

$s + C_{1} = \frac{8}{6} \int u^{3} d u$

Step 2.3.5.2

Cancel the common factor of $8$ and $6$ .

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Step 2.3.5.2.1

Factor $2$ out of $8$ .

$s + C_{1} = \frac{2 (4)}{6} \int u^{3} d u$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $2$ out of $6$ .

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

Step 2.3.5.2.2.2

Cancel the common factor.

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

Step 2.3.5.2.2.3

Rewrite the expression.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Multiply $3$ by $4$ .

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

Step 2.3.7.2.3

Cancel the common factor of $4$ and $12$ .

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Step 2.3.7.2.3.1

Factor $4$ out of $4$ .

$s + C_{1} = \frac{4 (1)}{12} u^{4} + C_{2}$

Step 2.3.7.2.3.2

Cancel the common factors.

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Step 2.3.7.2.3.2.1

Factor $4$ out of $12$ .

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

Step 2.3.7.2.3.2.2

Cancel the common factor.

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

Step 2.3.7.2.3.2.3

Rewrite the expression.

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

Step 2.3.8

Replace all occurrences of $u$ with $3 t^{2} - 5$ .

$s + C_{1} = \frac{1}{3} {(3 t^{2} - 5)}^{4} + C_{2}$

Step 2.4

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

$s = \frac{1}{3} {(3 t^{2} - 5)}^{4} + K$