Calculus Examples

$\frac{d r}{d s} = 3 r^{2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{r^{2}}$ .

$\frac{1}{r^{2}} \frac{d r}{d s} = \frac{1}{r^{2}} (3 r^{2})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{r^{2}} \frac{d r}{d s} = 3 \frac{1}{r^{2}} r^{2}$

Step 1.2.2

Combine $3$ and $\frac{1}{r^{2}}$ .

$\frac{1}{r^{2}} \frac{d r}{d s} = \frac{3}{r^{2}} r^{2}$

Step 1.2.3

Cancel the common factor of $r^{2}$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{1}{r^{2}} \frac{d r}{d s} = \frac{3}{r^{2}} r^{2}$

Step 1.2.3.2

Rewrite the expression.

$\frac{1}{r^{2}} \frac{d r}{d s} = 3$

Step 1.3

Rewrite the equation.

$\frac{1}{r^{2}} d r = 3 d s$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{r^{2}} d r = \int 3 d s$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

$\int {(r^{2})}^{- 1} d r = \int 3 d s$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$\int r^{2 \cdot - 1} d r = \int 3 d s$

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

$\int r^{- 2} d r = \int 3 d s$

Step 2.2.2

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

$- r^{- 1} + C_{1} = \int 3 d s$

Step 2.2.3

Rewrite $- r^{- 1} + C_{1}$ as $- \frac{1}{r} + C_{1}$ .

$- \frac{1}{r} + C_{1} = \int 3 d s$

Step 2.3

Apply the constant rule.

$- \frac{1}{r} + C_{1} = 3 s + C_{2}$

Step 2.4

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

$- \frac{1}{r} = 3 s + K$

Step 3

Solve for $r$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$r$

Step 3.2

Multiply each term in $- \frac{1}{r} = 3 s + K$ by $r$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{r} = 3 s + K$ by $r$ .

$- \frac{1}{r} r = 3 s r + K r$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $r$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{r}$ into the numerator.

$\frac{- 1}{r} r = 3 s r + K r$

Step 3.2.2.1.2

Cancel the common factor.

$\frac{- 1}{r} r = 3 s r + K r$

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = 3 s r + K r$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $3 s r + K r = - 1$ .

$3 s r + K r = - 1$

Step 3.3.2

Factor $r$ out of $3 s r + K r$ .

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Step 3.3.2.1

Factor $r$ out of $3 s r$ .

$r (3 s) + K r = - 1$

Step 3.3.2.2

Factor $r$ out of $K r$ .

$r (3 s) + r K = - 1$

Step 3.3.2.3

Factor $r$ out of $r (3 s) + r K$ .

$r (3 s + K) = - 1$

Step 3.3.3

Divide each term in $r (3 s + K) = - 1$ by $3 s + K$ and simplify.

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Step 3.3.3.1

Divide each term in $r (3 s + K) = - 1$ by $3 s + K$ .

$\frac{r (3 s + K)}{3 s + K} = \frac{- 1}{3 s + K}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $3 s + K$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{r (3 s + K)}{3 s + K} = \frac{- 1}{3 s + K}$

Step 3.3.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 1}{3 s + K}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$r = - \frac{1}{3 s + K}$