Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

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

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $- 8$ and $\frac{1}{2}$ .

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

Step 2.3.6.2.2

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.6.2.2.1

Factor $2$ out of $- 8$ .

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

Step 2.3.6.2.2.2

Cancel the common factors.

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Step 2.3.6.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.6.2.2.2.2

Cancel the common factor.

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

Step 2.3.6.2.2.2.3

Rewrite the expression.

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

Step 2.3.6.2.2.2.4

Divide $- 4$ by $1$ .

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

Step 2.3.6.2.3

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

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

Step 2.3.6.2.4

Cancel the common factor of $3$ .

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Step 2.3.6.2.4.1

Cancel the common factor.

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

Step 2.3.6.2.4.2

Rewrite the expression.

$y + C_{1} = - 4 x^{2} + 1 x^{3} + C_{4}$

Step 2.3.6.2.5

Multiply $x^{3}$ by $1$ .

$y + C_{1} = - 4 x^{2} + x^{3} + C_{4}$

Step 2.3.7

Reorder terms.

$y + C_{1} = x^{3} - 4 x^{2} + C_{4}$

Step 2.4

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

$y = x^{3} - 4 x^{2} + K$