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

$D$ .

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