Calculus Examples

$\frac{d y}{d t} = e^{- y} \sin (\frac{t}{3})$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{e^{- y}}$ .

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{1}{e^{- y}} (e^{- y} \sin (\frac{t}{3}))$

Step 1.2

Cancel the common factor of $e^{- y}$ .

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Step 1.2.1

Factor $e^{- y}$ out of $e^{- y} \sin (\frac{t}{3})$ .

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{1}{e^{- y}} (e^{- y} (\sin (\frac{t}{3})))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{1}{e^{- y}} (e^{- y} \sin (\frac{t}{3}))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{e^{- y}} \frac{d y}{d t} = \sin (\frac{t}{3})$

Step 1.3

Rewrite the equation.

$\frac{1}{e^{- y}} d y = \sin (\frac{t}{3}) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{- y}} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Negate the exponent of $e^{- y}$ and move it out of the denominator.

$\int 1 {(e^{- y})}^{- 1} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

Multiply the exponents in ${(e^{- y})}^{- 1}$ .

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Step 2.2.1.2.1.1

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

$\int 1 e^{- y \cdot - 1} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2.1.2.1.2

Multiply $- y \cdot - 1$ .

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Step 2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 y} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2.1.2.1.2.2

Multiply $y$ by $1$ .

$\int 1 e^{y} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2.1.2.2

Multiply $e^{y}$ by $1$ .

$\int e^{y} d y = \int \sin (\frac{t}{3}) d t$

Step 2.2.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$e^{y} + C_{1} = \int \sin (\frac{t}{3}) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $\frac{t}{3}$ .

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

Step 2.3.1.1.2

Since $\frac{1}{3}$ is constant with respect to $t$ , the derivative of $\frac{t}{3}$ with respect to $t$ is $\frac{1}{3} \frac{d}{d t} [t]$ .

$\frac{1}{3} \frac{d}{d t} [t]$

Step 2.3.1.1.3

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

$\frac{1}{3} \cdot 1$

Step 2.3.1.1.4

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

$\frac{1}{3}$

Step 2.3.1.2

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

$e^{y} + C_{1} = \int \sin (u) \frac{1}{\frac{1}{3}} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$e^{y} + C_{1} = \int \sin (u) (1 \cdot 3) d u$

Step 2.3.2.2

Multiply $3$ by $1$ .

$e^{y} + C_{1} = \int \sin (u) \cdot 3 d u$

Step 2.3.2.3

Move $3$ to the left of $\sin (u)$ .

$e^{y} + C_{1} = \int 3 \sin (u) d u$

Step 2.3.3

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$e^{y} + C_{1} = 3 \int \sin (u) d u$

Step 2.3.4

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$e^{y} + C_{1} = 3 (- \cos (u) + C_{2})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

$e^{y} + C_{1} = 3 (- \cos (u)) + C_{2}$

Step 2.3.5.2

Multiply $- 1$ by $3$ .

$e^{y} + C_{1} = - 3 \cos (u) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $\frac{t}{3}$ .

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

$e^{y} = - 3 \cos (\frac{1}{3} t) + K$

Step 3

Solve for $y$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (- 3 \cos (\frac{t}{3}) + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (- 3 \cos (\frac{t}{3}) + K)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (- 3 \cos (\frac{t}{3}) + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (- 3 \cos (\frac{t}{3}) + K)$