Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

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

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

Step 2.3.5.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.7.2.2

Multiply $x^{- 1}$ by $1$ .

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

Step 2.4

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

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