Calculus Examples

$3 x (y - 2) d x + (x^{2} + 1) d y = 0$

Step 1

Subtract $3 x (y - 2) d x$ from both sides of the equation.

$(x^{2} + 1) d y = - 3 x (y - 2) d x$

Step 2

Multiply both sides by $\frac{1}{(x^{2} + 1) (y - 2)}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{2} + 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Combine $- 3$ and $\frac{1}{(x^{2} + 1) (y - 2)}$ .

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

Step 3.4

Cancel the common factor of $y - 2$ .

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Step 3.4.1

Factor $y - 2$ out of $(x^{2} + 1) (y - 2)$ .

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

Step 3.4.2

Factor $y - 2$ out of $x (y - 2)$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 3.5

Combine $\frac{- 3}{x^{2} + 1}$ and $x$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Let $u_{1} = y - 2$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.2.1.1

Let $u_{1} = y - 2$ . Find $\frac{d u_{1}}{d y}$ .

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Step 4.2.1.1.1

Differentiate $y - 2$ .

$\frac{d}{d y} [y - 2]$

Step 4.2.1.1.2

By the Sum Rule, the derivative of $y - 2$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 2]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 2]$

Step 4.2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [- 2]$

Step 4.2.1.1.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

$1 + 0$

Step 4.2.1.1.5

Add $1$ and $0$ .

$1$

Step 4.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} d u_{1} = \int - \frac{3 x}{x^{2} + 1} d x$

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u_{1}$ with $y - 2$ .

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4.3.3

Multiply $3$ by $- 1$ .

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

Step 4.3.4

Let $u_{2} = x^{2} + 1$ . Then $d u_{2} = 2 x d x$ , so $\frac{1}{2} d u_{2} = x d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.3.4.1

Let $u_{2} = x^{2} + 1$ . Find $\frac{d u_{2}}{d x}$ .

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Step 4.3.4.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 4.3.4.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 4.3.4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 4.3.4.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 4.3.4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.3.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\ln (| y - 2 |) + C_{1} = - 3 \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 4.3.5

Simplify.

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Step 4.3.5.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$\ln (| y - 2 |) + C_{1} = - 3 \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 4.3.5.2

Move $2$ to the left of $u_{2}$ .

$\ln (| y - 2 |) + C_{1} = - 3 \int \frac{1}{2 u_{2}} d u_{2}$

Step 4.3.6

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\ln (| y - 2 |) + C_{1} = - 3 (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2})$

Step 4.3.7

Simplify.

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Step 4.3.7.1

Combine $\frac{1}{2}$ and $- 3$ .

$\ln (| y - 2 |) + C_{1} = \frac{- 3}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 4.3.7.2

Move the negative in front of the fraction.

$\ln (| y - 2 |) + C_{1} = - \frac{3}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 4.3.8

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| y - 2 |) + C_{1} = - \frac{3}{2} (\ln (| u_{2} |) + C_{2})$

Step 4.3.9

Simplify.

$\ln (| y - 2 |) + C_{1} = - \frac{3}{2} \ln (| u_{2} |) + C_{2}$

Step 4.3.10

Replace all occurrences of $u_{2}$ with $x^{2} + 1$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Combine $\ln (| x^{2} + 1 |)$ and $\frac{3}{2}$ .

$\ln (| y - 2 |) = - \frac{\ln (| x^{2} + 1 |) \cdot 3}{2} + C$

Step 5.1.1.2

Move $3$ to the left of $\ln (| x^{2} + 1 |)$ .

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

Step 5.2

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

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

Step 5.3

To write $\ln (| y - 2 |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (| y - 2 |) \cdot \frac{2}{2} + \frac{3 \ln (| x^{2} + 1 |)}{2} = C$

Step 5.4

Simplify terms.

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Step 5.4.1

Combine $\ln (| y - 2 |)$ and $\frac{2}{2}$ .

$\frac{\ln (| y - 2 |) \cdot 2}{2} + \frac{3 \ln (| x^{2} + 1 |)}{2} = C$

Step 5.4.2

Combine the numerators over the common denominator.

$\frac{\ln (| y - 2 |) \cdot 2 + 3 \ln (| x^{2} + 1 |)}{2} = C$

Step 5.5

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

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

Step 5.6

Simplify the left side.

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Step 5.6.1

Simplify $\frac{2 \ln (| y - 2 |) + 3 \ln (| x^{2} + 1 |)}{2}$ .

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Step 5.6.1.1

Simplify the numerator.

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Step 5.6.1.1.1

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

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

Step 5.6.1.1.2

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

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

Step 5.6.1.1.3

Simplify $3 \ln (| x^{2} + 1 |)$ by moving $3$ inside the logarithm.

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

Step 5.6.1.1.4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 5.6.1.2

Rewrite $\frac{\ln ({(y - 2)}^{2} {| x^{2} + 1 |}^{3})}{2}$ as $\frac{1}{2} \ln ({(y - 2)}^{2} {| x^{2} + 1 |}^{3})$ .

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

Step 5.6.1.3

Simplify $\frac{1}{2} \ln ({(y - 2)}^{2} {| x^{2} + 1 |}^{3})$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 5.6.1.4

Apply the product rule to ${(y - 2)}^{2} {| x^{2} + 1 |}^{3}$ .

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

Step 5.6.1.5

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

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Step 5.6.1.5.1

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

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

Step 5.6.1.5.2

Cancel the common factor of $2$ .

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Step 5.6.1.5.2.1

Cancel the common factor.

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

Step 5.6.1.5.2.2

Rewrite the expression.

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

Step 5.6.1.6

Simplify.

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

Step 5.6.1.7

Multiply the exponents in ${({| x^{2} + 1 |}^{3})}^{\frac{1}{2}}$ .

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Step 5.6.1.7.1

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

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

Step 5.6.1.7.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 5.6.1.8

Apply the distributive property.

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

Step 5.7

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y {| x^{2} + 1 |}^{\frac{3}{2}} - 2 {| x^{2} + 1 |}^{\frac{3}{2}})} = e^{C}$

Step 5.8

Rewrite $\ln (y {| x^{2} + 1 |}^{\frac{3}{2}} - 2 {| x^{2} + 1 |}^{\frac{3}{2}}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = y {| x^{2} + 1 |}^{\frac{3}{2}} - 2 {| x^{2} + 1 |}^{\frac{3}{2}}$

Step 5.9

Solve for $y$ .

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Step 5.9.1

Rewrite the equation as $y {| x^{2} + 1 |}^{\frac{3}{2}} - 2 {| x^{2} + 1 |}^{\frac{3}{2}} = e^{C}$ .

$y {| x^{2} + 1 |}^{\frac{3}{2}} - 2 {| x^{2} + 1 |}^{\frac{3}{2}} = e^{C}$

Step 5.9.2

Add $2 {| x^{2} + 1 |}^{\frac{3}{2}}$ to both sides of the equation.

$y {| x^{2} + 1 |}^{\frac{3}{2}} = e^{C} + 2 {| x^{2} + 1 |}^{\frac{3}{2}}$

Step 5.9.3

Divide each term in $y {| x^{2} + 1 |}^{\frac{3}{2}} = e^{C} + 2 {| x^{2} + 1 |}^{\frac{3}{2}}$ by ${| x^{2} + 1 |}^{\frac{3}{2}}$ and simplify.

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Step 5.9.3.1

Divide each term in $y {| x^{2} + 1 |}^{\frac{3}{2}} = e^{C} + 2 {| x^{2} + 1 |}^{\frac{3}{2}}$ by ${| x^{2} + 1 |}^{\frac{3}{2}}$ .

$\frac{y {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}} = \frac{e^{C}}{{| x^{2} + 1 |}^{\frac{3}{2}}} + \frac{2 {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}}$

Step 5.9.3.2

Simplify the left side.

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Step 5.9.3.2.1

Cancel the common factor.

$\frac{y {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}} = \frac{e^{C}}{{| x^{2} + 1 |}^{\frac{3}{2}}} + \frac{2 {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}}$

Step 5.9.3.2.2

Divide $y$ by $1$ .

$y = \frac{e^{C}}{{| x^{2} + 1 |}^{\frac{3}{2}}} + \frac{2 {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}}$

Step 5.9.3.3

Simplify the right side.

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Step 5.9.3.3.1

Combine the numerators over the common denominator.

$y = \frac{e^{C} + 2 {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}}$

Step 6

Simplify the constant of integration.

$y = \frac{C + 2 {| x^{2} + 1 |}^{\frac{3}{2}}}{{| x^{2} + 1 |}^{\frac{3}{2}}}$