Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $\frac{1}{y^{2}}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $y^{2}$ .

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Step 1.3.1.1

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

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

Step 1.3.1.2

Cancel the common factor.

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

Step 1.3.1.3

Rewrite the expression.

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

Step 1.3.2

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 1.3.3

Combine and simplify the denominator.

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Step 1.3.3.1

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 1.3.3.2

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 1.3.3.3

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Add $1$ and $1$ .

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

Step 1.3.3.6

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 1.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x^{2}}$ as ${(1 + x^{2})}^{\frac{1}{2}}$ .

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

Step 1.3.3.6.2

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

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

Step 1.3.3.6.3

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

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

Step 1.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.3.6.4.1

Cancel the common factor.

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

Step 1.3.3.6.4.2

Rewrite the expression.

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

Step 1.3.3.6.5

Simplify.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Rewrite $- y^{- 1} + C_{1}$ as $- \frac{1}{y} + C_{1}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $1 + x^{2}$ .

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

Step 2.3.1.1.2

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 2.3.1.1.3

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 2.3.1.1.4

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 2.3.1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.3.1.2

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

$- \frac{1}{y} + C_{1} = \int \frac{\sqrt{u}}{u} \cdot \frac{1}{2} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $\frac{\sqrt{u}}{u}$ by $\frac{1}{2}$ .

$- \frac{1}{y} + C_{1} = \int \frac{\sqrt{u}}{u \cdot 2} d u$

Step 2.3.2.2

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

$- \frac{1}{y} + C_{1} = \int \frac{\sqrt{u}}{2 u} d u$

Step 2.3.3

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

$- \frac{1}{y} + C_{1} = \frac{1}{2} \int \frac{\sqrt{u}}{u} d u$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Move $u^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2.3.4.2.2

Multiply $u$ by $u^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.4.2.2.1

Multiply $u$ by $u^{- \frac{1}{2}}$ .

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Step 2.3.4.2.2.1.1

Raise $u$ to the power of $1$ .

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

Step 2.3.4.2.2.1.2

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

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

Step 2.3.4.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.4.2.2.3

Combine the numerators over the common denominator.

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

Step 2.3.4.2.2.4

Subtract $1$ from $2$ .

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

Step 2.3.4.3

Apply basic rules of exponents.

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Step 2.3.4.3.1

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

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

Step 2.3.4.3.2

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

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Step 2.3.4.3.2.1

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

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

Step 2.3.4.3.2.2

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

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

Step 2.3.4.3.2.3

Move the negative in front of the fraction.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$- \frac{1}{y} + C_{1} = \frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C_{2}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

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

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

Step 2.3.6.2.2

Cancel the common factor of $2$ .

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Step 2.3.6.2.2.1

Cancel the common factor.

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

Step 2.3.6.2.2.2

Rewrite the expression.

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

Step 2.3.6.2.3

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

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

Step 2.3.7

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y$

Step 3.2

Multiply each term in $- \frac{1}{y} = {(1 + x^{2})}^{\frac{1}{2}} + K$ by $y$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{y} = {(1 + x^{2})}^{\frac{1}{2}} + K$ by $y$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Reorder factors in ${(1 + x^{2})}^{\frac{1}{2}} y + K y$ .

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

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $y {(1 + x^{2})}^{\frac{1}{2}} + K y = - 1$ .

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Factor $y$ out of $K y$ .

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

Step 3.3.2.3

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

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

Step 3.3.3

Divide each term in $y ({(1 + x^{2})}^{\frac{1}{2}} + K) = - 1$ by ${(1 + x^{2})}^{\frac{1}{2}} + K$ and simplify.

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Step 3.3.3.1

Divide each term in $y ({(1 + x^{2})}^{\frac{1}{2}} + K) = - 1$ by ${(1 + x^{2})}^{\frac{1}{2}} + K$ .

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Divide $y$ by $1$ .

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

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