Calculus Examples

$\frac{d y}{d x} = 4 \cos (e^{2 x}) \sin (e^{2 x}) e^{2 x}$

Step 1

Rewrite the equation.

$d y = 4 \cos (e^{2 x}) \sin (e^{2 x}) e^{2 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 \cos (e^{2 x}) \sin (e^{2 x}) e^{2 x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 \cos (e^{2 x}) \sin (e^{2 x}) e^{2 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int \cos (e^{2 x}) \sin (e^{2 x}) e^{2 x} d x$

Step 2.3.2

Let $u_{3} = \cos (e^{2 x})$ . Then $d u_{3} = - 2 e^{2 x} \sin (e^{2 x}) d x$ , so $- \frac{1}{2} d u_{3} = e^{2 x} \sin (e^{2 x}) d x$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 2.3.2.1

Let $u_{3} = \cos (e^{2 x})$ . Find $\frac{d u_{3}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $\cos (e^{2 x})$ .

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

Step 2.3.2.1.2

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

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Step 2.3.2.1.2.1

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

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

Step 2.3.2.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d x} [e^{2 x}]$

Step 2.3.2.1.2.3

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

$- \sin (e^{2 x}) \frac{d}{d x} [e^{2 x}]$

Step 2.3.2.1.3

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 $g (x) = 2 x$ .

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Step 2.3.2.1.3.1

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

$- \sin (e^{2 x}) (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [2 x])$

Step 2.3.2.1.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- \sin (e^{2 x}) (e^{u_{2}} \frac{d}{d x} [2 x])$

Step 2.3.2.1.3.3

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

$- \sin (e^{2 x}) (e^{2 x} \frac{d}{d x} [2 x])$

Step 2.3.2.1.4

Differentiate.

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Step 2.3.2.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- \sin (e^{2 x}) (e^{2 x} (2 \frac{d}{d x} [x]))$

Step 2.3.2.1.4.2

Multiply $2$ by $- 1$ .

$- 2 \sin (e^{2 x}) (e^{2 x} (\frac{d}{d x} [x]))$

Step 2.3.2.1.4.3

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

$- 2 \sin (e^{2 x}) (e^{2 x} \cdot 1)$

Step 2.3.2.1.4.4

Simplify the expression.

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Step 2.3.2.1.4.4.1

Multiply $e^{2 x}$ by $1$ .

$- 2 \sin (e^{2 x}) e^{2 x}$

Step 2.3.2.1.4.4.2

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

$- 2 e^{2 x} \sin (e^{2 x})$

Step 2.3.2.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$y + C_{1} = 4 \int u_{3} \frac{1}{- 2} d u_{3}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Move the negative in front of the fraction.

$y + C_{1} = 4 \int u_{3} (- \frac{1}{2}) d u_{3}$

Step 2.3.3.2

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

$y + C_{1} = 4 \int - \frac{u_{3}}{2} d u_{3}$

Step 2.3.4

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

$y + C_{1} = 4 (- \int \frac{u_{3}}{2} d u_{3})$

Step 2.3.5

Multiply $- 1$ by $4$ .

$y + C_{1} = - 4 \int \frac{u_{3}}{2} d u_{3}$

Step 2.3.6

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = - 4 (\frac{1}{2} \int u_{3} d u_{3})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{2}$ and $- 4$ .

$y + C_{1} = \frac{- 4}{2} \int u_{3} d u_{3}$

Step 2.3.7.2

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.7.2.1

Factor $2$ out of $- 4$ .

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

Step 2.3.7.2.2

Cancel the common factors.

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Step 2.3.7.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.7.2.2.2

Cancel the common factor.

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

Step 2.3.7.2.2.3

Rewrite the expression.

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

Step 2.3.7.2.2.4

Divide $- 2$ by $1$ .

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

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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Step 2.3.9.1

Rewrite $- 2 (\frac{1}{2} {u_{3}}^{2} + C_{2})$ as $- 2 (\frac{1}{2}) {u_{3}}^{2} + C_{2}$ .

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

Step 2.3.9.2

Simplify.

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Step 2.3.9.2.1

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

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

Step 2.3.9.2.2

Cancel the common factor of $- 2$ and $2$ .

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Step 2.3.9.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.9.2.2.2

Cancel the common factors.

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Step 2.3.9.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.9.2.2.2.2

Cancel the common factor.

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

Step 2.3.9.2.2.2.3

Rewrite the expression.

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

Step 2.3.9.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.3.10

Replace all occurrences of $u_{3}$ with $\cos (e^{2 x})$ .

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

Step 2.4

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

$y = - \cos^{2} (e^{2 x}) + K$