Calculus Examples

$\frac{d s}{d t} = 13 t {(5 t^{2} - 1)}^{3}$

Step 1

Rewrite the equation.

$d s = 13 t {(5 t^{2} - 1)}^{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 13 t {(5 t^{2} - 1)}^{3} d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 13 t {(5 t^{2} - 1)}^{3} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $13$ is constant with respect to $t$ , move $13$ out of the integral.

$s + C_{1} = 13 \int t {(5 t^{2} - 1)}^{3} d t$

Step 2.3.2

Let $u = 5 t^{2} - 1$ . Then $d u = 10 t d t$ , so $\frac{1}{10} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 5 t^{2} - 1$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $5 t^{2} - 1$ .

$\frac{d}{d t} [5 t^{2} - 1]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $5 t^{2} - 1$ with respect to $t$ is $\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [- 1]$ .

$\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [- 1]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d t} [5 t^{2}]$ .

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Step 2.3.2.1.3.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{2}$ with respect to $t$ is $5 \frac{d}{d t} [t^{2}]$ .

$5 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 1]$

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 $n = 2$ .

$5 (2 t) + \frac{d}{d t} [- 1]$

Step 2.3.2.1.3.3

Multiply $2$ by $5$ .

$10 t + \frac{d}{d t} [- 1]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

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

$10 t + 0$

Step 2.3.2.1.4.2

Add $10 t$ and $0$ .

$10 t$

Step 2.3.2.2

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

$s + C_{1} = 13 \int u^{3} \frac{1}{10} d u$

Step 2.3.3

Combine $u^{3}$ and $\frac{1}{10}$ .

$s + C_{1} = 13 \int \frac{u^{3}}{10} d u$

Step 2.3.4

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$s + C_{1} = 13 (\frac{1}{10} \int u^{3} d u)$

Step 2.3.5

Combine $\frac{1}{10}$ and $13$ .

$s + C_{1} = \frac{13}{10} \int u^{3} d u$

Step 2.3.6

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

$s + C_{1} = \frac{13}{10} (\frac{1}{4} u^{4} + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite $\frac{13}{10} (\frac{1}{4} u^{4} + C_{2})$ as $\frac{13}{10} \cdot \frac{1}{4} u^{4} + C_{2}$ .

$s + C_{1} = \frac{13}{10} \cdot \frac{1}{4} u^{4} + C_{2}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $\frac{13}{10}$ by $\frac{1}{4}$ .

$s + C_{1} = \frac{13}{10 \cdot 4} u^{4} + C_{2}$

Step 2.3.7.2.2

Multiply $10$ by $4$ .

$s + C_{1} = \frac{13}{40} u^{4} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $5 t^{2} - 1$ .

$s + C_{1} = \frac{13}{40} {(5 t^{2} - 1)}^{4} + C_{2}$

Step 2.4

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

$s = \frac{13}{40} {(5 t^{2} - 1)}^{4} + K$