Calculus Examples

$\frac{d y}{d x} = (- \frac{6}{x^{2}} + \frac{1}{6})$

Step 1

Rewrite the equation.

$d y = (- \frac{6}{x^{2}} + \frac{1}{6}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int (- \frac{6}{x^{2}} + \frac{1}{6}) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int (- \frac{6}{x^{2}} + \frac{1}{6}) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int - \frac{6}{x^{2}} d x + \int \frac{1}{6} d x$

Step 2.3.2

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

$y + C_{1} = - \int \frac{6}{x^{2}} d x + \int \frac{1}{6} d x$

Step 2.3.3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$y + C_{1} = - (6 \int \frac{1}{x^{2}} d x) + \int \frac{1}{6} d x$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Multiply $6$ by $- 1$ .

$y + C_{1} = - 6 \int \frac{1}{x^{2}} d x + \int \frac{1}{6} d x$

Step 2.3.4.2

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{1} = - 6 \int {(x^{2})}^{- 1} d x + \int \frac{1}{6} d x$

Step 2.3.4.3

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{1} = - 6 \int x^{2 \cdot - 1} d x + \int \frac{1}{6} d x$

Step 2.3.4.3.2

Multiply $2$ by $- 1$ .

$y + C_{1} = - 6 \int x^{- 2} d x + \int \frac{1}{6} d x$

Step 2.3.5

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

$y + C_{1} = - 6 (- x^{- 1} + C_{2}) + \int \frac{1}{6} d x$

Step 2.3.6

Apply the constant rule.

$y + C_{1} = - 6 (- x^{- 1} + C_{2}) + \frac{1}{6} x + C_{3}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$y + C_{1} = - 6 (- \frac{1}{x}) + \frac{1}{6} x + C_{4}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $- 1$ by $- 6$ .

$y + C_{1} = 6 \frac{1}{x} + \frac{1}{6} x + C_{4}$

Step 2.3.7.2.2

Combine $6$ and $\frac{1}{x}$ .

$y + C_{1} = \frac{6}{x} + \frac{1}{6} x + C_{4}$

Step 2.3.8

Reorder terms.

$y + C_{1} = \frac{1}{6} x + \frac{6}{x} + C_{4}$

Step 2.4

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

$y = \frac{1}{6} x + \frac{6}{x} + K$