Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.3.6

Reorder terms.

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

Step 2.4

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

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