Calculus Examples

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

Step 1

Rewrite the equation.

$d y = (8 - 2 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 - 2 x d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.5.2.2.2

Cancel the common factors.

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Step 2.3.5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.5.2.2.2.2

Cancel the common factor.

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

Step 2.3.5.2.2.2.3

Rewrite the expression.

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

Step 2.3.5.2.2.2.4

Divide $- 1$ by $1$ .

$y + C_{1} = 8 x - x^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$y + C_{1} = - x^{2} + 8 x + C_{4}$

Step 2.4

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

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