Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $- x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$- 1 \cdot 1$

Step 1.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 1.2

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

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

Step 2

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

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

Step 3

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 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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 3.2

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

$- \int \tan (u_{2}) d u_{2}$

Step 4

The integral of $\tan (u_{2})$ with respect to $u_{2}$ is $\ln (| \sec (u_{2}) |)$ .

$- (\ln (| \sec (u_{2}) |) + C)$

Step 5

Simplify.

$- \ln (| \sec (u_{2}) |) + C$

Step 6

Substitute back in for each integration substitution variable.

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Step 6.1

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

$- \ln (| \sec (e^{u_{1}}) |) + C$

Step 6.2

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

$- \ln (| \sec (e^{- x}) |) + C$