Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of ${(1 + x^{2})}^{\frac{1}{2}}$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Cancel the common factor.

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

Step 3.1.4

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of ${(1 + y^{2})}^{\frac{1}{2}}$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

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

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Differentiate $1 + y^{2}$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.1.3

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

$0 + \frac{d}{d y} [y^{2}]$

Step 4.2.1.1.4

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

$0 + 2 y$

Step 4.2.1.1.5

Add $0$ and $2 y$ .

$2 y$

Step 4.2.1.2

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Move $2$ to the left of ${u_{1}}^{\frac{1}{2}}$ .

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

Step 4.2.3

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

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

Step 4.2.4

Apply basic rules of exponents.

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Step 4.2.4.1

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

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

Step 4.2.4.2

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

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Step 4.2.4.2.1

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

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

Step 4.2.4.2.2

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

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

Step 4.2.4.2.3

Move the negative in front of the fraction.

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

Step 4.2.5

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

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

Step 4.2.6

Simplify.

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Step 4.2.6.1

Rewrite $\frac{1}{2} (2 {u_{1}}^{\frac{1}{2}} + C_{1})$ as $\frac{1}{2} \cdot 2 {u_{1}}^{\frac{1}{2}} + C_{1}$ .

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

Step 4.2.6.2

Simplify.

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Step 4.2.6.2.1

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

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

Step 4.2.6.2.2

Cancel the common factor of $2$ .

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Step 4.2.6.2.2.1

Cancel the common factor.

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

Step 4.2.6.2.2.2

Rewrite the expression.

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

Step 4.2.6.2.3

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

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

Step 4.2.7

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Let $u_{2} = 1 + x^{2}$ . 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 4.3.1.1

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

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Step 4.3.1.1.1

Differentiate $1 + x^{2}$ .

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

Step 4.3.1.1.2

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

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

Step 4.3.1.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 4.3.1.1.4

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

$0 + 2 x$

Step 4.3.1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 4.3.1.2

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

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Move $2$ to the left of ${u_{2}}^{\frac{1}{2}}$ .

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

Step 4.3.3

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

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

Step 4.3.4

Apply basic rules of exponents.

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Step 4.3.4.1

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

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

Step 4.3.4.2

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

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Step 4.3.4.2.1

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

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

Step 4.3.4.2.2

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

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

Step 4.3.4.2.3

Move the negative in front of the fraction.

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

Step 4.3.5

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

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = \frac{1}{2} (2 {u_{2}}^{\frac{1}{2}} + C_{2})$

Step 4.3.6

Simplify.

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Step 4.3.6.1

Rewrite $\frac{1}{2} (2 {u_{2}}^{\frac{1}{2}} + C_{2})$ as $\frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C_{2}$ .

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = \frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 4.3.6.2

Simplify.

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Step 4.3.6.2.1

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

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = \frac{2}{2} {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 4.3.6.2.2

Cancel the common factor of $2$ .

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Step 4.3.6.2.2.1

Cancel the common factor.

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = \frac{2}{2} {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 4.3.6.2.2.2

Rewrite the expression.

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = 1 {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 4.3.6.2.3

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

${(1 + y^{2})}^{\frac{1}{2}} + C_{1} = {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 4.3.7

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

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

Step 4.4

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

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