Calculus Examples

$\frac{d P}{d C} = \frac{- 216}{{(C + 1)}^{\frac{3}{2}}}$

Step 1

Rewrite the equation.

$d P = \frac{- 216}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d P = \int \frac{- 216}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2.2

Apply the constant rule.

$P + C_{1} = \int \frac{- 216}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2.3

Integrate the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$P + C_{1} = \int - \frac{216}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2.3.2

Since $- 1$ is constant with respect to $C$ , move $- 1$ out of the integral.

$P + C_{1} = - \int \frac{216}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2.3.3

Since $216$ is constant with respect to $C$ , move $216$ out of the integral.

$P + C_{1} = - (216 \int \frac{1}{{(C + 1)}^{\frac{3}{2}}} d C)$

Step 2.3.4

Multiply $216$ by $- 1$ .

$P + C_{1} = - 216 \int \frac{1}{{(C + 1)}^{\frac{3}{2}}} d C$

Step 2.3.5

Let $u = C + 1$ . Then $d u = d C$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.5.1

Let $u = C + 1$ . Find $\frac{d u}{d C}$ .

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Step 2.3.5.1.1

Differentiate $C + 1$ .

$\frac{d}{d C} [C + 1]$

Step 2.3.5.1.2

By the Sum Rule, the derivative of $C + 1$ with respect to $C$ is $\frac{d}{d C} [C] + \frac{d}{d C} [1]$ .

$\frac{d}{d C} [C] + \frac{d}{d C} [1]$

Step 2.3.5.1.3

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

$1 + \frac{d}{d C} [1]$

Step 2.3.5.1.4

Since $1$ is constant with respect to $C$ , the derivative of $1$ with respect to $C$ is $0$ .

$1 + 0$

Step 2.3.5.1.5

Add $1$ and $0$ .

$1$

Step 2.3.5.2

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

$P + C_{1} = - 216 \int \frac{1}{u^{\frac{3}{2}}} d u$

Step 2.3.6

Apply basic rules of exponents.

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Step 2.3.6.1

Move $u^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

$P + C_{1} = - 216 \int {(u^{\frac{3}{2}})}^{- 1} d u$

Step 2.3.6.2

Multiply the exponents in ${(u^{\frac{3}{2}})}^{- 1}$ .

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Step 2.3.6.2.1

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

$P + C_{1} = - 216 \int u^{\frac{3}{2} \cdot - 1} d u$

Step 2.3.6.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 2.3.6.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

$P + C_{1} = - 216 \int u^{\frac{3 \cdot - 1}{2}} d u$

Step 2.3.6.2.2.2

Multiply $3$ by $- 1$ .

$P + C_{1} = - 216 \int u^{\frac{- 3}{2}} d u$

Step 2.3.6.2.3

Move the negative in front of the fraction.

$P + C_{1} = - 216 \int u^{- \frac{3}{2}} d u$

Step 2.3.7

By the Power Rule, the integral of $u^{- \frac{3}{2}}$ with respect to $u$ is $- 2 u^{- \frac{1}{2}}$ .

$P + C_{1} = - 216 (- 2 u^{- \frac{1}{2}} + C_{2})$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Rewrite $- 216 (- 2 u^{- \frac{1}{2}} + C_{2})$ as $- 216 (- 2 \frac{1}{u^{\frac{1}{2}}}) + C_{2}$ .

$P + C_{1} = - 216 (- 2 \frac{1}{u^{\frac{1}{2}}}) + C_{2}$

Step 2.3.8.2

Simplify.

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Step 2.3.8.2.1

Combine $- 2$ and $\frac{1}{u^{\frac{1}{2}}}$ .

$P + C_{1} = - 216 \frac{- 2}{u^{\frac{1}{2}}} + C_{2}$

Step 2.3.8.2.2

Move the negative in front of the fraction.

$P + C_{1} = - 216 (- \frac{2}{u^{\frac{1}{2}}}) + C_{2}$

Step 2.3.8.2.3

Multiply $- 1$ by $- 216$ .

$P + C_{1} = 216 \frac{2}{u^{\frac{1}{2}}} + C_{2}$

Step 2.3.8.2.4

Combine $216$ and $\frac{2}{u^{\frac{1}{2}}}$ .

$P + C_{1} = \frac{216 \cdot 2}{u^{\frac{1}{2}}} + C_{2}$

Step 2.3.8.2.5

Multiply $216$ by $2$ .

$P + C_{1} = \frac{432}{u^{\frac{1}{2}}} + C_{2}$

Step 2.3.9

Replace all occurrences of $u$ with $C + 1$ .

$P + C_{1} = \frac{432}{{(C + 1)}^{\frac{1}{2}}} + C_{2}$

Step 2.4

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

$P = \frac{432}{{(C + 1)}^{\frac{1}{2}}} + K$