Calculus Examples

$\frac{d^{2} s}{d t^{2}} = \sin (3 t) + \cos (3 t)$

Step 1

Integrate both sides with respect to $t$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $t$ .

$\frac{d s}{d t} = \int \sin (3 t) + \cos (3 t) d t$

Step 1.2

Split the single integral into multiple integrals.

$\frac{d s}{d t} = \int \sin (3 t) d t + \int \cos (3 t) d t$

Step 1.3

Let $u_{1} = 3 t$ . Then $d u_{1} = 3 d t$ , so $\frac{1}{3} d u_{1} = d t$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.3.1

Let $u_{1} = 3 t$ . Find $\frac{d u_{1}}{d t}$ .

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Step 1.3.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 1.3.1.2

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

$3 \frac{d}{d t} [t]$

Step 1.3.1.3

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

$3 \cdot 1$

Step 1.3.1.4

Multiply $3$ by $1$ .

$3$

Step 1.3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{d s}{d t} = \int \sin (u_{1}) \frac{1}{3} d u_{1} + \int \cos (3 t) d t$

Step 1.4

Combine $\sin (u_{1})$ and $\frac{1}{3}$ .

$\frac{d s}{d t} = \int \frac{\sin (u_{1})}{3} d u_{1} + \int \cos (3 t) d t$

Step 1.5

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

$\frac{d s}{d t} = \frac{1}{3} \int \sin (u_{1}) d u_{1} + \int \cos (3 t) d t$

Step 1.6

The integral of $\sin (u_{1})$ with respect to $u_{1}$ is $- \cos (u_{1})$ .

$\frac{d s}{d t} = \frac{1}{3} (- \cos (u_{1}) + C_{1}) + \int \cos (3 t) d t$

Step 1.7

Let $u_{2} = 3 t$ . Then $d u_{2} = 3 d t$ , so $\frac{1}{3} d u_{2} = d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.7.1

Let $u_{2} = 3 t$ . Find $\frac{d u_{2}}{d t}$ .

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Step 1.7.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 1.7.1.2

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

$3 \frac{d}{d t} [t]$

Step 1.7.1.3

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

$3 \cdot 1$

Step 1.7.1.4

Multiply $3$ by $1$ .

$3$

Step 1.7.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{d s}{d t} = \frac{1}{3} (- \cos (u_{1}) + C_{1}) + \int \cos (u_{2}) \frac{1}{3} d u_{2}$

Step 1.8

Combine $\cos (u_{2})$ and $\frac{1}{3}$ .

$\frac{d s}{d t} = \frac{1}{3} (- \cos (u_{1}) + C_{1}) + \int \frac{\cos (u_{2})}{3} d u_{2}$

Step 1.9

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$\frac{d s}{d t} = \frac{1}{3} (- \cos (u_{1}) + C_{1}) + \frac{1}{3} \int \cos (u_{2}) d u_{2}$

Step 1.10

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{d s}{d t} = \frac{1}{3} (- \cos (u_{1}) + C_{1}) + \frac{1}{3} (\sin (u_{2}) + C_{2})$

Step 1.11

Simplify.

$\frac{d s}{d t} = - \frac{\cos (u_{1})}{3} + \frac{1}{3} \sin (u_{2}) + C_{3}$

Step 1.12

Substitute back in for each integration substitution variable.

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Step 1.12.1

Replace all occurrences of $u_{1}$ with $3 t$ .

$\frac{d s}{d t} = - \frac{\cos (3 t)}{3} + \frac{1}{3} \sin (u_{2}) + C_{3}$

Step 1.12.2

Replace all occurrences of $u_{2}$ with $3 t$ .

$\frac{d s}{d t} = - \frac{\cos (3 t)}{3} + \frac{1}{3} \sin (3 t) + C_{3}$

Step 1.13

Reorder terms.

$\frac{d s}{d t} = - \frac{1}{3} \cos (3 t) + \frac{1}{3} \sin (3 t) + C_{3}$

Step 2

Rewrite the equation.

$d s = (- \frac{1}{3} \cos (3 t) + \frac{1}{3} \sin (3 t) + C_{3}) d t$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d s = \int - \frac{1}{3} \cos (3 t) + \frac{1}{3} \sin (3 t) + C_{3} d t$

Step 3.2

Apply the constant rule.

$s + C_{4} = \int - \frac{1}{3} \cos (3 t) + \frac{1}{3} \sin (3 t) + C_{3} d t$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$s + C_{4} = \int - \frac{1}{3} \cos (3 t) d t + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.2

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

$s + C_{4} = - \frac{1}{3} \int \cos (3 t) d t + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.3

Let $u_{3} = 3 t$ . Then $d u_{3} = 3 d t$ , so $\frac{1}{3} d u_{3} = d t$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 3.3.3.1

Let $u_{3} = 3 t$ . Find $\frac{d u_{3}}{d t}$ .

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Step 3.3.3.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 3.3.3.1.2

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

$3 \frac{d}{d t} [t]$

Step 3.3.3.1.3

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

$3 \cdot 1$

Step 3.3.3.1.4

Multiply $3$ by $1$ .

$3$

Step 3.3.3.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$s + C_{4} = - \frac{1}{3} \int \cos (u_{3}) \frac{1}{3} d u_{3} + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.4

Combine $\cos (u_{3})$ and $\frac{1}{3}$ .

$s + C_{4} = - \frac{1}{3} \int \frac{\cos (u_{3})}{3} d u_{3} + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.5

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

$s + C_{4} = - \frac{1}{3} (\frac{1}{3} \int \cos (u_{3}) d u_{3}) + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.6

Simplify.

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Step 3.3.6.1

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

$s + C_{4} = - \frac{1}{3 \cdot 3} \int \cos (u_{3}) d u_{3} + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.6.2

Multiply $3$ by $3$ .

$s + C_{4} = - \frac{1}{9} \int \cos (u_{3}) d u_{3} + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.7

The integral of $\cos (u_{3})$ with respect to $u_{3}$ is $\sin (u_{3})$ .

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \int \frac{1}{3} \sin (3 t) d t + \int C_{3} d t$

Step 3.3.8

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

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{3} \int \sin (3 t) d t + \int C_{3} d t$

Step 3.3.9

Let $u_{4} = 3 t$ . Then $d u_{4} = 3 d t$ , so $\frac{1}{3} d u_{4} = d t$ . Rewrite using $u_{4}$ and $d$ $u_{4}$ .

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Step 3.3.9.1

Let $u_{4} = 3 t$ . Find $\frac{d u_{4}}{d t}$ .

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Step 3.3.9.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 3.3.9.1.2

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

$3 \frac{d}{d t} [t]$

Step 3.3.9.1.3

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

$3 \cdot 1$

Step 3.3.9.1.4

Multiply $3$ by $1$ .

$3$

Step 3.3.9.2

Rewrite the problem using $u_{4}$ and $d u_{4}$ .

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{3} \int \sin (u_{4}) \frac{1}{3} d u_{4} + \int C_{3} d t$

Step 3.3.10

Combine $\sin (u_{4})$ and $\frac{1}{3}$ .

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{3} \int \frac{\sin (u_{4})}{3} d u_{4} + \int C_{3} d t$

Step 3.3.11

Since $\frac{1}{3}$ is constant with respect to $u_{4}$ , move $\frac{1}{3}$ out of the integral.

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{3} (\frac{1}{3} \int \sin (u_{4}) d u_{4}) + \int C_{3} d t$

Step 3.3.12

Simplify.

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Step 3.3.12.1

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

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{3 \cdot 3} \int \sin (u_{4}) d u_{4} + \int C_{3} d t$

Step 3.3.12.2

Multiply $3$ by $3$ .

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{9} \int \sin (u_{4}) d u_{4} + \int C_{3} d t$

Step 3.3.13

The integral of $\sin (u_{4})$ with respect to $u_{4}$ is $- \cos (u_{4})$ .

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{9} (- \cos (u_{4}) + C_{6}) + \int C_{3} d t$

Step 3.3.14

Apply the constant rule.

$s + C_{4} = - \frac{1}{9} (\sin (u_{3}) + C_{5}) + \frac{1}{9} (- \cos (u_{4}) + C_{6}) + C_{3} t + C_{7}$

Step 3.3.15

Simplify.

$s + C_{4} = - \frac{\sin (u_{3})}{9} - \frac{\cos (u_{4})}{9} + C_{3} t + C_{8}$

Step 3.3.16

Substitute back in for each integration substitution variable.

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Step 3.3.16.1

Replace all occurrences of $u_{3}$ with $3 t$ .

$s + C_{4} = - \frac{\sin (3 t)}{9} - \frac{\cos (u_{4})}{9} + C_{3} t + C_{8}$

Step 3.3.16.2

Replace all occurrences of $u_{4}$ with $3 t$ .

$s + C_{4} = - \frac{\sin (3 t)}{9} - \frac{\cos (3 t)}{9} + C_{3} t + C_{8}$

Step 3.3.17

Reorder terms.

$s + C_{4} = - \frac{1}{9} \sin (3 t) - \frac{1}{9} \cos (3 t) + C_{3} t + C_{8}$

Step 3.4

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

$s = - \frac{1}{9} \sin (3 t) - \frac{1}{9} \cos (3 t) + C t + D$