Calculus Examples

$(e^{x} + 1) d x + \frac{y^{2} - 1}{y^{2}} d y = 0$

Step 1

Subtract $(e^{x} + 1) d x$ from both sides of the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

Multiply $(y^{2} - 1) y^{- 2}$ .

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

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

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Step 2.2.3.1.1

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

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

Step 2.2.3.1.2

Subtract $2$ from $2$ .

$\int y^{0} - 1 y^{- 2} d y = \int - (e^{x} + 1) d x$

Step 2.2.3.2

Simplify $y^{0}$ .

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

Step 2.2.3.3

Rewrite $- 1 y^{- 2}$ as $- y^{- 2}$ .

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

Step 2.2.4

Split the single integral into multiple integrals.

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

Step 2.2.5

Apply the constant rule.

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify.

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Step 2.2.8.1

Simplify.

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

Step 2.2.8.2

Simplify.

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Step 2.2.8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.8.2.2

Multiply $\frac{1}{y}$ by $1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 2.3.4

Apply the constant rule.

$y + \frac{1}{y} + C_{3} = - (e^{x} + C_{4} + x + C_{5})$

Step 2.3.5

Simplify.

$y + \frac{1}{y} + C_{3} = - (e^{x} + x) + C_{6}$

Step 2.4

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

$y + \frac{1}{y} = - (e^{x} + x) + K$