Calculus Examples

$\frac{d y}{d x} = \frac{1}{x^{3}} - x$

Step 1

Rewrite the equation.

$d y = (\frac{1}{x^{3}} - x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{1}{x^{3}} - x d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

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

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{2}$ .

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

Step 2.3.6.2.2

Move $2$ to the left of $x^{2}$ .

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

Step 2.3.6.2.3

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 2.3.7

Reorder terms.

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

Step 2.3.8

Reorder terms.

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

Step 2.4

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

$y = - \frac{1}{2} x^{2} - \frac{1}{2 x^{2}} + K$