Calculus Examples

$\int e^{- 3 x} \sec (e^{- 3 x}) \tan (e^{- 3 x}) d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $- 3 x$ .

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

Step 1.1.2

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 1.1.3

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

$- 3 \cdot 1$

Step 1.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 1.2

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

$\int e^{u_{1}} \sec (e^{u_{1}}) \tan (e^{u_{1}}) \frac{1}{- 3} d u_{1}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int e^{u_{1}} \sec (e^{u_{1}}) \tan (e^{u_{1}}) (- \frac{1}{3}) d u_{1}$

Step 2.2

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

$\int \sec (e^{u_{1}}) \tan (e^{u_{1}}) (- \frac{e^{u_{1}}}{3}) d u_{1}$

Step 2.3

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

$\int \tan (e^{u_{1}}) (- \frac{\sec (e^{u_{1}}) e^{u_{1}}}{3}) d u_{1}$

Step 2.4

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

$\int - \frac{\tan (e^{u_{1}}) \sec (e^{u_{1}}) e^{u_{1}}}{3} d u_{1}$

Step 3

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

$- \int \frac{\tan (e^{u_{1}}) \sec (e^{u_{1}}) e^{u_{1}}}{3} d u_{1}$

Step 4

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

$- (\frac{1}{3} \int \tan (e^{u_{1}}) \sec (e^{u_{1}}) e^{u_{1}} d u_{1})$

Step 5

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 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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 5.2

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

$- \frac{1}{3} \int \tan (u_{2}) \sec (u_{2}) d u_{2}$

Step 6

Since the derivative of $\sec (u_{2})$ is $\tan (u_{2}) \sec (u_{2})$ , the integral of $\tan (u_{2}) \sec (u_{2})$ is $\sec (u_{2})$ .

$- \frac{1}{3} (\sec (u_{2}) + C)$

Step 7

Simplify.

$- \frac{1}{3} \sec (u_{2}) + C$

Step 8

Substitute back in for each integration substitution variable.

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Step 8.1

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

$- \frac{1}{3} \sec (e^{u_{1}}) + C$

Step 8.2

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

$- \frac{1}{3} \sec (e^{- 3 x}) + C$