Calculus Examples

$\int 6 e^{6 x} \sin (e^{6 x}) d x$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int e^{6 x} \sin (e^{6 x}) d x$

Step 2

Let $u_{1} = 6 x$ . Then $d u_{1} = 6 d x$ , so $\frac{1}{6} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 6 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 2.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 2.1.3

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

$6 \cdot 1$

Step 2.1.4

Multiply $6$ by $1$ .

$6$

Step 2.2

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

$6 \int e^{u_{1}} \sin (e^{u_{1}}) \frac{1}{6} d u_{1}$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{6}$ and $e^{u_{1}}$ .

$6 \int \frac{e^{u_{1}}}{6} \sin (e^{u_{1}}) d u_{1}$

Step 3.2

Combine $\frac{e^{u_{1}}}{6}$ and $\sin (e^{u_{1}})$ .

$6 \int \frac{e^{u_{1}} \sin (e^{u_{1}})}{6} d u_{1}$

Step 4

Since $\frac{1}{6}$ is constant with respect to $u_{1}$ , move $\frac{1}{6}$ out of the integral.

$6 (\frac{1}{6} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{6}$ and $6$ .

$\frac{6}{6} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1}$

Step 5.2

Cancel the common factor of $6$ .

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Step 5.2.1

Cancel the common factor.

$\frac{6}{6} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1}$

Step 5.2.2

Rewrite the expression.

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

Step 5.3

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $e^{u_{1}}$ .

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

Step 6.1.2

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

$e^{u_{1}}$

Step 6.2

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

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

Step 7

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

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

Step 8

Substitute back in for each integration substitution variable.

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Step 8.1

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

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

Step 8.2

Replace all occurrences of $u_{1}$ with $6 x$ .

$- \cos (e^{6 x}) + C$