Calculus Examples

$\frac{d R}{d t} = \frac{60}{t^{2}}$ , $R (1) = 20$

Step 1

Rewrite the equation.

$d R = \frac{60}{t^{2}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d R = \int \frac{60}{t^{2}} d t$

Step 2.2

Apply the constant rule.

$R + C_{1} = \int \frac{60}{t^{2}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $60$ is constant with respect to $t$ , move $60$ out of the integral.

$R + C_{1} = 60 \int \frac{1}{t^{2}} d t$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

Move $t^{2}$ out of the denominator by raising it to the $- 1$ power.

$R + C_{1} = 60 \int {(t^{2})}^{- 1} d t$

Step 2.3.2.2

Multiply the exponents in ${(t^{2})}^{- 1}$ .

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Step 2.3.2.2.1

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

$R + C_{1} = 60 \int t^{2 \cdot - 1} d t$

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

$R + C_{1} = 60 \int t^{- 2} d t$

Step 2.3.3

By the Power Rule, the integral of $t^{- 2}$ with respect to $t$ is $- t^{- 1}$ .

$R + C_{1} = 60 (- t^{- 1} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $60 (- t^{- 1} + C_{2})$ as $60 (- \frac{1}{t}) + C_{2}$ .

$R + C_{1} = 60 (- \frac{1}{t}) + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $- 1$ by $60$ .

$R + C_{1} = - 60 \frac{1}{t} + C_{2}$

Step 2.3.4.2.2

Combine $- 60$ and $\frac{1}{t}$ .

$R + C_{1} = \frac{- 60}{t} + C_{2}$

Step 2.3.4.2.3

Move the negative in front of the fraction.

$R + C_{1} = - \frac{60}{t} + C_{2}$

Step 2.4

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

$R = - \frac{60}{t} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $20$ for $R$ in $R = - \frac{60}{t} + K$ .

$20 = - \frac{60}{1} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{60}{1} + K = 20$ .

$- \frac{60}{1} + K = 20$

Step 4.2

Simplify each term.

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Step 4.2.1

Divide $60$ by $1$ .

$- 1 \cdot 60 + K = 20$

Step 4.2.2

Multiply $- 1$ by $60$ .

$- 60 + K = 20$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Add $60$ to both sides of the equation.

$K = 20 + 60$

Step 4.3.2

Add $20$ and $60$ .

$K = 80$

Step 5

Substitute $80$ for $K$ in $R = - \frac{60}{t} + K$ and simplify.

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Step 5.1

Substitute $80$ for $K$ .

$R = - \frac{60}{t} + 80$