Calculus Examples

$\frac{d r}{d q} = 178 - q - 1.5 q^{2}$

Step 1

Rewrite the equation.

$d r = (178 - q - 1.5 q^{2}) d q$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d r = \int 178 - q - 1.5 q^{2} d q$

Step 2.2

Apply the constant rule.

$r + C_{1} = \int 178 - q - 1.5 q^{2} d q$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$r + C_{1} = \int 178 d q + \int - q d q + \int - 1.5 q^{2} d q$

Step 2.3.2

Apply the constant rule.

$r + C_{1} = 178 q + C_{2} + \int - q d q + \int - 1.5 q^{2} d q$

Step 2.3.3

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

$r + C_{1} = 178 q + C_{2} - \int q d q + \int - 1.5 q^{2} d q$

Step 2.3.4

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

$r + C_{1} = 178 q + C_{2} - (\frac{1}{2} q^{2} + C_{3}) + \int - 1.5 q^{2} d q$

Step 2.3.5

Since $- 1.5$ is constant with respect to $q$ , move $- 1.5$ out of the integral.

$r + C_{1} = 178 q + C_{2} - (\frac{1}{2} q^{2} + C_{3}) - 1.5 \int q^{2} d q$

Step 2.3.6

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

$r + C_{1} = 178 q + C_{2} - (\frac{1}{2} q^{2} + C_{3}) - 1.5 (\frac{1}{3} q^{3} + C_{4})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$r + C_{1} = 178 q - \frac{1}{2} q^{2} - 1.5 (\frac{1}{3}) q^{3} + C_{5}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Combine $- 1.5$ and $\frac{1}{3}$ .

$r + C_{1} = 178 q - \frac{1}{2} q^{2} + \frac{- 1.5}{3} q^{3} + C_{5}$

Step 2.3.7.2.2

Move the negative in front of the fraction.

$r + C_{1} = 178 q - \frac{1}{2} q^{2} - \frac{1.5}{3} q^{3} + C_{5}$

Step 2.3.7.3

Simplify.

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Step 2.3.7.3.1

Divide $1.5$ by $3$ .

$r + C_{1} = 178 q - \frac{1}{2} q^{2} - 1 \cdot 0.5 q^{3} + C_{5}$

Step 2.3.7.3.2

Multiply $- 1$ by $0.5$ .

$r + C_{1} = 178 q - \frac{1}{2} q^{2} - 0.5 q^{3} + C_{5}$

Step 2.3.7.3.3

Combine $q^{2}$ and $\frac{1}{2}$ .

$r + C_{1} = 178 q - \frac{q^{2}}{2} - 0.5 q^{3} + C_{5}$

Step 2.3.8

Reorder terms.

$r + C_{1} = 178 q - \frac{1}{2} q^{2} - 0.5 q^{3} + C_{5}$

Step 2.3.9

Reorder terms.

$r + C_{1} = - 0.5 q^{3} - \frac{1}{2} q^{2} + 178 q + C_{5}$

Step 2.4

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

$r = - 0.5 q^{3} - \frac{1}{2} q^{2} + 178 q + K$