Calculus Examples

$\int e^{2 x} \sin (e^{x}) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$\int e^{u_{1}} \sin (e^{\frac{u_{1}}{2}}) \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{e^{u_{1}}}{2} \sin (e^{\frac{u_{1}}{2}}) d u_{1}$

Step 2.2

Combine $\frac{e^{u_{1}}}{2}$ and $\sin (e^{\frac{u_{1}}{2}})$ .

$\int \frac{e^{u_{1}} \sin (e^{\frac{u_{1}}{2}})}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int e^{u_{1}} \sin (e^{\frac{u_{1}}{2}}) d u_{1}$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\frac{u_{1}}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$

Step 4.1.2

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$

Step 4.1.3

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

$\frac{1}{2} \cdot 1$

Step 4.1.4

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

$\frac{1}{2}$

Step 4.2

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

$\frac{1}{2} \int e^{2 u_{2}} \sin (e^{u_{2}}) \frac{1}{\frac{1}{2}} d u_{2}$

Step 5

Simplify.

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Step 5.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{2} \int e^{2 u_{2}} \sin (e^{u_{2}}) (1 \cdot 2) d u_{2}$

Step 5.2

Multiply $2$ by $1$ .

$\frac{1}{2} \int e^{2 u_{2}} \sin (e^{u_{2}}) \cdot 2 d u_{2}$

Step 5.3

Move $2$ to the left of $e^{2 u_{2}} \sin (e^{u_{2}})$ .

$\frac{1}{2} \int 2 e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$

Step 6

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

$\frac{1}{2} (2 \int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{2}{2} \int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

$\frac{2}{2} \int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$

Step 7.2.2

Rewrite the expression.

$1 \int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$

Step 7.3

Multiply $\int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$ by $1$ .

$\int e^{2 u_{2}} \sin (e^{u_{2}}) d u_{2}$

Step 8

Let $u_{3} = e^{u_{2}}$ . Then $d u_{3} = e^{u_{2}} d u_{2}$ , so $\frac{1}{e^{u_{2}}} d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 8.1

Let $u_{3} = e^{u_{2}}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 8.1.1

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

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

Step 8.1.2

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

$e^{u_{2}}$

Step 8.2

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

$\int u_{3} \sin (u_{3}) d u_{3}$

Step 9

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u_{3}$ and $dv = \sin (u_{3})$ .

$u_{3} (- \cos (u_{3})) - \int - \cos (u_{3}) d u_{3}$

Step 10

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

$u_{3} (- \cos (u_{3})) - - \int \cos (u_{3}) d u_{3}$

Step 11

Simplify.

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Step 11.1

Multiply $- 1$ by $- 1$ .

$u_{3} (- \cos (u_{3})) + 1 \int \cos (u_{3}) d u_{3}$

Step 11.2

Multiply $\int \cos (u_{3}) d u_{3}$ by $1$ .

$u_{3} (- \cos (u_{3})) + \int \cos (u_{3}) d u_{3}$

Step 12

The integral of $\cos (u_{3})$ with respect to $u_{3}$ is $\sin (u_{3})$ .

$u_{3} (- \cos (u_{3})) + \sin (u_{3}) + C$

Step 13

Rewrite $u_{3} (- \cos (u_{3})) + \sin (u_{3}) + C$ as $- u_{3} \cos (u_{3}) + \sin (u_{3}) + C$ .

$- u_{3} \cos (u_{3}) + \sin (u_{3}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{3}$ with $e^{u_{2}}$ .

$- e^{u_{2}} \cos (e^{u_{2}}) + \sin (e^{u_{2}}) + C$

Step 14.2

Replace all occurrences of $u_{2}$ with $\frac{u_{1}}{2}$ .

$- e^{\frac{u_{1}}{2}} \cos (e^{\frac{u_{1}}{2}}) + \sin (e^{\frac{u_{1}}{2}}) + C$

Step 14.3

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

$- e^{\frac{2 x}{2}} \cos (e^{\frac{2 x}{2}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15

Simplify.

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Step 15.1

Reduce the expression $\frac{2 x}{2}$ by cancelling the common factors.

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Step 15.1.1

Cancel the common factor.

$- e^{\frac{2 x}{2}} \cos (e^{\frac{2 x}{2}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15.1.2

Rewrite the expression.

$- e^{\frac{x}{1}} \cos (e^{\frac{2 x}{2}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15.2

Reduce the expression $\frac{2 x}{2}$ by cancelling the common factors.

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Step 15.2.1

Cancel the common factor.

$- e^{\frac{x}{1}} \cos (e^{\frac{2 x}{2}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15.2.2

Rewrite the expression.

$- e^{\frac{x}{1}} \cos (e^{\frac{x}{1}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15.3

Reduce the expression $\frac{2 x}{2}$ by cancelling the common factors.

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Step 15.3.1

Cancel the common factor.

$- e^{\frac{x}{1}} \cos (e^{\frac{x}{1}}) + \sin (e^{\frac{2 x}{2}}) + C$

Step 15.3.2

Rewrite the expression.

$- e^{\frac{x}{1}} \cos (e^{\frac{x}{1}}) + \sin (e^{\frac{x}{1}}) + C$

Step 15.4

Divide $x$ by $1$ .

$- e^{x} \cos (e^{\frac{x}{1}}) + \sin (e^{\frac{x}{1}}) + C$

Step 15.5

Divide $x$ by $1$ .

$- e^{x} \cos (e^{x}) + \sin (e^{\frac{x}{1}}) + C$

Step 15.6

Divide $x$ by $1$ .

$- e^{x} \cos (e^{x}) + \sin (e^{x}) + C$