Calculus Examples

$x (y + 2) d y = (\ln (x) + 1) d x$

Step 1

Multiply both sides by $\frac{1}{x}$ .

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

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Multiply $\frac{1}{x}$ by $\ln (x) + 1$ .

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Integrate the left side.

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Step 3.2.1

Split the single integral into multiple integrals.

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

Step 3.2.2

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

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

Step 3.2.3

Apply the constant rule.

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

Step 3.2.4

Simplify.

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the fraction into multiple fractions.

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

Step 3.3.2

Split the single integral into multiple integrals.

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

Step 3.3.3

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.3.3.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 3.3.3.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 3.3.3.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 3.3.3.2

Rewrite the problem using $u$ and $d u$ .

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

Step 3.3.4

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

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

Step 3.3.5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} y^{2} + 2 y + C_{3} = \frac{1}{2} u^{2} + C_{4} + \ln (| x |) + C_{5}$

Step 3.3.6

Simplify.

$\frac{1}{2} y^{2} + 2 y + C_{3} = \frac{1}{2} u^{2} + \ln (| x |) + C_{6}$

Step 3.3.7

Replace all occurrences of $u$ with $\ln (x)$ .

$\frac{1}{2} y^{2} + 2 y + C_{3} = \frac{1}{2} \ln^{2} (x) + \ln (| x |) + C_{6}$

Step 3.4

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

$\frac{1}{2} y^{2} + 2 y = \frac{1}{2} \ln^{2} (x) + \ln (| x |) + C$

Step 4

Solve for $y$ .

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Step 4.1

Simplify the expressions in the equation.

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Step 4.1.1

Simplify the left side.

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Step 4.1.1.1

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

$\frac{y^{2}}{2} + 2 y = \frac{1}{2} \ln^{2} (x) + \ln (| x |) + C$

Step 4.1.2

Simplify the right side.

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Step 4.1.2.1

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

$\frac{y^{2}}{2} + 2 y = \frac{\ln^{2} (x)}{2} + \ln (| x |) + C$

Step 4.2

Multiply each term in $\frac{y^{2}}{2} + 2 y = \frac{\ln^{2} (x)}{2} + \ln (| x |) + C$ by $2$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{y^{2}}{2} + 2 y = \frac{\ln^{2} (x)}{2} + \ln (| x |) + C$ by $2$ .

$\frac{y^{2}}{2} \cdot 2 + 2 y \cdot 2 = \frac{\ln^{2} (x)}{2} \cdot 2 + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.1

Cancel the common factor.

$\frac{y^{2}}{2} \cdot 2 + 2 y \cdot 2 = \frac{\ln^{2} (x)}{2} \cdot 2 + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.2.1.1.2

Rewrite the expression.

$y^{2} + 2 y \cdot 2 = \frac{\ln^{2} (x)}{2} \cdot 2 + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.2.1.2

Multiply $2$ by $2$ .

$y^{2} + 4 y = \frac{\ln^{2} (x)}{2} \cdot 2 + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.1

Cancel the common factor.

$y^{2} + 4 y = \frac{\ln^{2} (x)}{2} \cdot 2 + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.3.1.1.2

Rewrite the expression.

$y^{2} + 4 y = \ln^{2} (x) + \ln (| x |) \cdot 2 + C \cdot 2$

Step 4.2.3.1.2

Move $2$ to the left of $\ln (| x |)$ .

$y^{2} + 4 y = \ln^{2} (x) + 2 \cdot \ln (| x |) + C \cdot 2$

Step 4.2.3.1.3

Move $2$ to the left of $C$ .

$y^{2} + 4 y = \ln^{2} (x) + 2 \ln (| x |) + 2 C$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$y^{2} + 4 y = \ln^{2} (x) + \ln ({| x |}^{2}) + 2 C$

Step 4.3.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$y^{2} + 4 y = \ln^{2} (x) + \ln (x^{2}) + 2 C$

Step 4.4

Move all the terms containing a logarithm to the left side of the equation.

$- \ln (x^{2}) = - y^{2} - 4 y + \ln^{2} (x) + 2 C$

Step 4.5

Rewrite the equation as $- y^{2} - 4 y + \ln^{2} (x) + 2 C = - \ln (x^{2})$ .

$- y^{2} - 4 y + \ln^{2} (x) + 2 C = - \ln (x^{2})$

Step 4.6

Add $\ln (x^{2})$ to both sides of the equation.

$- y^{2} - 4 y + \ln^{2} (x) + 2 C + \ln (x^{2}) = 0$

Step 4.7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.8

Substitute the values $a = - 1$ , $b = - 4$ , and $c = \ln^{2} (x) + 2 C + \ln (x^{2})$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2})))}}{2 \cdot - 1}$

Step 4.9

Simplify.

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Step 4.9.1

Simplify the numerator.

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Step 4.9.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2}))$ .

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Step 4.9.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2}))}}{2 \cdot - 1}$

Step 4.9.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2}))$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2})))}}{2 \cdot - 1}$

Step 4.9.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2})))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2})))}}{2 \cdot - 1}$

Step 4.9.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2}))$ .

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Step 4.9.1.2.1

Reorder $- 4$ and $- 1 \cdot (\ln^{2} (x) + 2 C + \ln (x^{2}))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (\ln^{2} (x) + 2 C + \ln (x^{2})) - 4)}}{2 \cdot - 1}$

Step 4.9.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (\ln^{2} (x) + 2 C + \ln (x^{2})) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 4.9.1.2.3

Factor $- 1$ out of $- 1 (\ln^{2} (x) + 2 C + \ln (x^{2})) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4))}}{2 \cdot - 1}$

Step 4.9.1.2.4

Rewrite $- 1 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4)$ as $- (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4))}}{2 \cdot - 1}$

$y = \frac{4 \pm \sqrt{- 4 \cdot (- 1 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4))}}{2 \cdot - 1}$

Step 4.9.1.3

Combine exponents.

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Step 4.9.1.3.1

Factor out negative.

$y = \frac{4 \pm \sqrt{- (- 4 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4))}}{2 \cdot - 1}$

Step 4.9.1.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 \pm \sqrt{4 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4)}}{2 \cdot - 1}$

Step 4.9.1.4

Rewrite $4 (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4)$ as $2^{2} (\ln^{2} (x) + 2 C + \ln (x^{2}) + 2^{2})$ .

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Step 4.9.1.4.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{4 \pm \sqrt{2^{2} (\ln^{2} (x) + 2 C + \ln (x^{2}) + 4)}}{2 \cdot - 1}$

Step 4.9.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{4 \pm \sqrt{2^{2} (\ln^{2} (x) + 2 C + \ln (x^{2}) + 2^{2})}}{2 \cdot - 1}$

Step 4.9.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 2^{2}}}{2 \cdot - 1}$

Step 4.9.1.6

Raise $2$ to the power of $2$ .

$y = \frac{4 \pm 2 \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}}{2 \cdot - 1}$

Step 4.9.2

Multiply $2$ by $- 1$ .

$y = \frac{4 \pm 2 \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}}{- 2}$

Step 4.9.3

Simplify $\frac{4 \pm 2 \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}}{- 2}$ .

$y = \frac{2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}}{- 1}$

Step 4.9.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}}{- 1}$ .

$y = - 1 \cdot (2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4})$

Step 4.9.5

Rewrite $- 1 \cdot (2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4})$ as $- (2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4})$ .

$y = - (2 \pm \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4})$

Step 4.10

The final answer is the combination of both solutions.

$y = - 2 - \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}$

$y = - 2 + \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}$

$y = - 2 - \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}$

$y = - 2 + \sqrt{\ln^{2} (x) + 2 C + \ln (x^{2}) + 4}$

Step 5

Simplify the constant of integration.

$y = - 2 - \sqrt{\ln^{2} (x) + C + \ln (x^{2}) + 4}$

$y = - 2 + \sqrt{\ln^{2} (x) + C + \ln (x^{2}) + 4}$