Calculus Examples

$\frac{d y}{d x} = \frac{9 x^{2} - 6}{x^{2}}$

Step 1

Rewrite the equation.

$d y = \frac{9 x^{2} - 6}{x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{9 x^{2} - 6}{x^{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{9 x^{2} - 6}{x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

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

$y + C_{1} = \int (9 x^{2} - 6) {(x^{2})}^{- 1} d x$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$y + C_{1} = \int (9 x^{2} - 6) x^{2 \cdot - 1} d x$

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

$y + C_{1} = \int (9 x^{2} - 6) x^{- 2} d x$

Step 2.3.2

Multiply $(9 x^{2} - 6) x^{- 2}$ .

$y + C_{1} = \int 9 x^{2} x^{- 2} - 6 x^{- 2} d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $x^{2}$ by $x^{- 2}$ by adding the exponents.

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Step 2.3.3.1.1

Move $x^{- 2}$ .

$y + C_{1} = \int 9 (x^{- 2} x^{2}) - 6 x^{- 2} d x$

Step 2.3.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int 9 x^{- 2 + 2} - 6 x^{- 2} d x$

Step 2.3.3.1.3

Add $- 2$ and $2$ .

$y + C_{1} = \int 9 x^{0} - 6 x^{- 2} d x$

Step 2.3.3.2

Simplify $9 x^{0}$ .

$y + C_{1} = \int 9 - 6 x^{- 2} d x$

Step 2.3.4

Split the single integral into multiple integrals.

$y + C_{1} = \int 9 d x + \int - 6 x^{- 2} d x$

Step 2.3.5

Apply the constant rule.

$y + C_{1} = 9 x + C_{2} + \int - 6 x^{- 2} d x$

Step 2.3.6

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

$y + C_{1} = 9 x + C_{2} - 6 \int x^{- 2} d x$

Step 2.3.7

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

$y + C_{1} = 9 x + C_{2} - 6 (- x^{- 1} + C_{3})$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Simplify.

$y + C_{1} = 9 x - 6 (- \frac{1}{x}) + C_{4}$

Step 2.3.8.2

Simplify.

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Step 2.3.8.2.1

Multiply $- 1$ by $- 6$ .

$y + C_{1} = 9 x + 6 \frac{1}{x} + C_{4}$

Step 2.3.8.2.2

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

$y + C_{1} = 9 x + \frac{6}{x} + C_{4}$

Step 2.4

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

$y = 9 x + \frac{6}{x} + K$