Calculus Examples

$3 e^{3 x} \sin (e^{3 x})$

Step 1

Write $3 e^{3 x} \sin (e^{3 x})$ as a function.

$f (x) = 3 e^{3 x} \sin (e^{3 x})$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 3 e^{3 x} \sin (e^{3 x}) d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int e^{3 x} \sin (e^{3 x}) d x$

Step 5

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

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Step 5.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.1.3

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

$3 \cdot 1$

Step 5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2

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

$3 \int e^{u_{1}} \sin (e^{u_{1}}) \frac{1}{3} d u_{1}$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $e^{u_{1}}$ .

$3 \int \frac{e^{u_{1}}}{3} \sin (e^{u_{1}}) d u_{1}$

Step 6.2

Combine $\frac{e^{u_{1}}}{3}$ and $\sin (e^{u_{1}})$ .

$3 \int \frac{e^{u_{1}} \sin (e^{u_{1}})}{3} d u_{1}$

Step 7

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

$3 (\frac{1}{3} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1})$

Step 8

Simplify.

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Step 8.1

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

$\frac{3}{3} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1}$

Step 8.2

Cancel the common factor of $3$ .

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Step 8.2.1

Cancel the common factor.

$\frac{3}{3} \int e^{u_{1}} \sin (e^{u_{1}}) d u_{1}$

Step 8.2.2

Rewrite the expression.

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

Step 8.3

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 9

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 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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 9.2

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

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

Step 10

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

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

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

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

Step 11.2

Replace all occurrences of $u_{1}$ with $3 x$ .

$- \cos (e^{3 x}) + C$

Step 12

The answer is the antiderivative of the function $f (x) = 3 e^{3 x} \sin (e^{3 x})$ .

$F (x) =$ $- \cos (e^{3 x}) + C$