Calculus Examples

$\frac{d y}{d x} = - \frac{1}{x^{2}} - \frac{3}{x^{4}} + 12$

Step 1

Rewrite the equation.

$d y = (- \frac{1}{x^{2}} - \frac{3}{x^{4}} + 12) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - \frac{1}{x^{2}} - \frac{3}{x^{4}} + 12 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - \frac{1}{x^{2}} - \frac{3}{x^{4}} + 12 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int - \frac{1}{x^{2}} d x + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.2

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

$y + C_{1} = - \int \frac{1}{x^{2}} d x + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.3

Apply basic rules of exponents.

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Step 2.3.3.1

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

$y + C_{1} = - \int {(x^{2})}^{- 1} d x + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.3.2

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

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Step 2.3.3.2.1

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

$y + C_{1} = - \int x^{2 \cdot - 1} d x + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.3.2.2

Multiply $2$ by $- 1$ .

$y + C_{1} = - \int x^{- 2} d x + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.4

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

$y + C_{1} = - (- x^{- 1} + C_{2}) + \int - \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.5

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

$y + C_{1} = - (- x^{- 1} + C_{2}) - \int \frac{3}{x^{4}} d x + \int 12 d x$

Step 2.3.6

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

$y + C_{1} = - (- x^{- 1} + C_{2}) - (3 \int \frac{1}{x^{4}} d x) + \int 12 d x$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Multiply $3$ by $- 1$ .

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 \int \frac{1}{x^{4}} d x + \int 12 d x$

Step 2.3.7.2

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 \int {(x^{4})}^{- 1} d x + \int 12 d x$

Step 2.3.7.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 2.3.7.3.1

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

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 \int x^{4 \cdot - 1} d x + \int 12 d x$

Step 2.3.7.3.2

Multiply $4$ by $- 1$ .

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 \int x^{- 4} d x + \int 12 d x$

Step 2.3.8

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 (- \frac{1}{3} x^{- 3} + C_{3}) + \int 12 d x$

Step 2.3.9

Simplify.

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Step 2.3.9.1

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

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 (- \frac{x^{- 3}}{3} + C_{3}) + \int 12 d x$

Step 2.3.9.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 (- \frac{1}{3 x^{3}} + C_{3}) + \int 12 d x$

Step 2.3.10

Apply the constant rule.

$y + C_{1} = - (- x^{- 1} + C_{2}) - 3 (- \frac{1}{3 x^{3}} + C_{3}) + 12 x + C_{4}$

Step 2.3.11

Simplify.

$y + C_{1} = \frac{1}{x} + \frac{1}{x^{3}} + 12 x + C_{5}$

Step 2.3.12

Reorder terms.

$y + C_{1} = 12 x + \frac{1}{x} + \frac{1}{x^{3}} + C_{5}$

Step 2.4

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

$y = 12 x + \frac{1}{x} + \frac{1}{x^{3}} + K$