Calculus Examples

$\frac{d y}{d x} = 8 + 2 x - 3 x^{2}$

Step 1

Rewrite the equation.

$d y = (8 + 2 x - 3 x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 + 2 x - 3 x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 + 2 x - 3 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 8 d x + \int 2 x d x + \int - 3 x^{2} d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = 8 x + C_{2} + \int 2 x d x + \int - 3 x^{2} d x$

Step 2.3.3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 8 x + C_{2} + 2 \int x d x + \int - 3 x^{2} d x$

Step 2.3.4

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

$y + C_{1} = 8 x + C_{2} + 2 (\frac{1}{2} x^{2} + C_{3}) + \int - 3 x^{2} d x$

Step 2.3.5

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

$y + C_{1} = 8 x + C_{2} + 2 (\frac{1}{2} x^{2} + C_{3}) - 3 \int x^{2} d x$

Step 2.3.6

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

$y + C_{1} = 8 x + C_{2} + 2 (\frac{1}{2} x^{2} + C_{3}) - 3 (\frac{1}{3} x^{3} + C_{4})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$y + C_{1} = 8 x + x^{2} - 3 (\frac{1}{3} x^{3}) + C_{5}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y + C_{1} = 8 x + x^{2} - 3 \frac{x^{3}}{3} + C_{5}$

Step 2.3.7.2.2

Combine $- 3$ and $\frac{x^{3}}{3}$ .

$y + C_{1} = 8 x + x^{2} + \frac{- 3 x^{3}}{3} + C_{5}$

Step 2.3.7.2.3

Cancel the common factor of $- 3$ and $3$ .

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Step 2.3.7.2.3.1

Factor $3$ out of $- 3 x^{3}$ .

$y + C_{1} = 8 x + x^{2} + \frac{3 (- x^{3})}{3} + C_{5}$

Step 2.3.7.2.3.2

Cancel the common factors.

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Step 2.3.7.2.3.2.1

Factor $3$ out of $3$ .

$y + C_{1} = 8 x + x^{2} + \frac{3 (- x^{3})}{3 (1)} + C_{5}$

Step 2.3.7.2.3.2.2

Cancel the common factor.

$y + C_{1} = 8 x + x^{2} + \frac{3 (- x^{3})}{3 \cdot 1} + C_{5}$

Step 2.3.7.2.3.2.3

Rewrite the expression.

$y + C_{1} = 8 x + x^{2} + \frac{- x^{3}}{1} + C_{5}$

Step 2.3.7.2.3.2.4

Divide $- x^{3}$ by $1$ .

$y + C_{1} = 8 x + x^{2} - x^{3} + C_{5}$

Step 2.3.8

Reorder terms.

$y + C_{1} = - x^{3} + x^{2} + 8 x + C_{5}$

Step 2.4

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

$y = - x^{3} + x^{2} + 8 x + K$