Calculus Examples

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

Step 1

Multiply both sides by $x y^{2}$ .

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

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Factor $x$ out of $x y^{2}$ .

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

Step 2.1.2

Cancel the common factor.

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

Step 2.1.3

Rewrite the expression.

$y^{2} d y = x y^{2} \frac{e^{x^{2}}}{y^{2}} d x$

Step 2.2

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

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Step 2.2.1

Factor $y^{2}$ out of $x y^{2}$ .

$y^{2} d y = y^{2} x \frac{e^{x^{2}}}{y^{2}} d x$

Step 2.2.2

Cancel the common factor.

$y^{2} d y = y^{2} x \frac{e^{x^{2}}}{y^{2}} d x$

Step 2.2.3

Rewrite the expression.

$y^{2} d y = x e^{x^{2}} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int y^{2} d y = \int x e^{x^{2}} d x$

Step 3.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

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

Step 3.3

Integrate the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Differentiate $e^{x^{2}}$ .

$\frac{d}{d x} [e^{x^{2}}]$

Step 3.3.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 3.3.1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 3.3.1.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.1.1.2.3

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

$e^{x^{2}} \frac{d}{d x} [x^{2}]$

Step 3.3.1.1.3

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

$e^{x^{2}} (2 x)$

Step 3.3.1.1.4

Simplify.

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Step 3.3.1.1.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

$2 e^{x^{2}} x$

Step 3.3.1.1.4.2

Reorder factors in $2 e^{x^{2}} x$ .

$2 x e^{x^{2}}$

Step 3.3.1.2

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

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

Step 3.3.2

Apply the constant rule.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{2} u_{2} + C_{2}$

Step 3.3.3

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

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

Step 3.4

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

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

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{2} e^{x^{2}} + K)$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

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Step 4.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{2} e^{x^{2}} + K)$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{2} e^{x^{2}} + K)$

Step 4.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{2} e^{x^{2}} + K)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $3 (\frac{1}{2} e^{x^{2}} + K)$ .

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Step 4.2.2.1.1

Combine $\frac{1}{2}$ and $e^{x^{2}}$ .

$y^{3} = 3 (\frac{e^{x^{2}}}{2} + K)$

Step 4.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{e^{x^{2}}}{2} + 3 K$

Step 4.2.2.1.3

Combine $3$ and $\frac{e^{x^{2}}}{2}$ .

$y^{3} = \frac{3 e^{x^{2}}}{2} + 3 K$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{\frac{3 e^{x^{2}}}{2} + 3 K}$

Step 4.4

Simplify $\sqrt[3]{\frac{3 e^{x^{2}}}{2} + 3 K}$ .

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Step 4.4.1

Factor $3$ out of $\frac{3 e^{x^{2}}}{2} + 3 K$ .

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Step 4.4.1.1

Factor $3$ out of $\frac{3 e^{x^{2}}}{2}$ .

$y = \sqrt[3]{3 \frac{e^{x^{2}}}{2} + 3 K}$

Step 4.4.1.2

Factor $3$ out of $3 K$ .

$y = \sqrt[3]{3 \frac{e^{x^{2}}}{2} + 3 K}$

Step 4.4.1.3

Factor $3$ out of $3 \frac{e^{x^{2}}}{2} + 3 K$ .

$y = \sqrt[3]{3 (\frac{e^{x^{2}}}{2} + K)}$

Step 4.4.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{3 (\frac{e^{x^{2}}}{2} + K \cdot \frac{2}{2})}$

Step 4.4.3

Simplify terms.

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Step 4.4.3.1

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

$y = \sqrt[3]{3 (\frac{e^{x^{2}}}{2} + \frac{K \cdot 2}{2})}$

Step 4.4.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 \frac{e^{x^{2}} + K \cdot 2}{2}}$

Step 4.4.4

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

$y = \sqrt[3]{3 \frac{e^{x^{2}} + 2 K}{2}}$

Step 4.4.5

Combine $3$ and $\frac{e^{x^{2}} + 2 K}{2}$ .

$y = \sqrt[3]{\frac{3 (e^{x^{2}} + 2 K)}{2}}$

Step 4.4.6

Rewrite $\sqrt[3]{\frac{3 (e^{x^{2}} + 2 K)}{2}}$ as $\frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)}}{\sqrt[3]{2}}$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)}}{\sqrt[3]{2}}$

Step 4.4.7

Multiply $\frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 4.4.8

Combine and simplify the denominator.

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Step 4.4.8.1

Multiply $\frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 4.4.8.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 4.4.8.3

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

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 4.4.8.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 4.4.8.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 4.4.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 4.4.8.5.2

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

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 4.4.8.5.3

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

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 4.4.8.5.4

Cancel the common factor of $3$ .

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Step 4.4.8.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 4.4.8.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 4.4.8.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} {\sqrt[3]{2}}^{2}}{2}$

Step 4.4.9

Simplify the numerator.

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Step 4.4.9.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} \sqrt[3]{2^{2}}}{2}$

Step 4.4.9.2

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

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K)} \sqrt[3]{4}}{2}$

Step 4.4.10

Simplify the numerator.

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Step 4.4.10.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{3 (e^{x^{2}} + 2 K) \cdot 4}}{2}$

Step 4.4.10.2

Multiply $4$ by $3$ .

$y = \frac{\sqrt[3]{12 (e^{x^{2}} + 2 K)}}{2}$

Step 5

Simplify the constant of integration.

$y = \frac{\sqrt[3]{12 (e^{x^{2}} + K)}}{2}$