Calculus Examples

$\frac{d z}{d h} = h^{2} - 3 h + 2$

Step 1

Rewrite the equation.

$d z = (h^{2} - 3 h + 2) d h$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d z = \int h^{2} - 3 h + 2 d h$

Step 2.2

Apply the constant rule.

$z + C_{1} = \int h^{2} - 3 h + 2 d h$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$z + C_{1} = \int h^{2} d h + \int - 3 h d h + \int 2 d h$

Step 2.3.2

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

$z + C_{1} = \frac{1}{3} h^{3} + C_{2} + \int - 3 h d h + \int 2 d h$

Step 2.3.3

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

$z + C_{1} = \frac{1}{3} h^{3} + C_{2} - 3 \int h d h + \int 2 d h$

Step 2.3.4

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

$z + C_{1} = \frac{1}{3} h^{3} + C_{2} - 3 (\frac{1}{2} h^{2} + C_{3}) + \int 2 d h$

Step 2.3.5

Apply the constant rule.

$z + C_{1} = \frac{1}{3} h^{3} + C_{2} - 3 (\frac{1}{2} h^{2} + C_{3}) + 2 h + C_{4}$

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$z + C_{1} = \frac{1}{3} h^{3} + C_{2} - 3 (\frac{h^{2}}{2} + C_{3}) + 2 h + C_{4}$

Step 2.3.6.2

Simplify.

$z + C_{1} = \frac{h^{3}}{3} - \frac{3 h^{2}}{2} + 2 h + C_{5}$

Step 2.3.7

Reorder terms.

$z + C_{1} = \frac{1}{3} h^{3} - \frac{3}{2} h^{2} + 2 h + C_{5}$

Step 2.4

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

$z = \frac{1}{3} h^{3} - \frac{3}{2} h^{2} + 2 h + K$