Calculus Examples

$\int e^{t} \cot (e^{t} - 4) d t$

Step 1

Let $u = e^{t} - 4$ . Then $d u = e^{t} d t$ , so $\frac{1}{e^{t}} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = e^{t} - 4$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $e^{t} - 4$ .

$\frac{d}{d t} [e^{t} - 4]$

Step 1.1.2

By the Sum Rule, the derivative of $e^{t} - 4$ with respect to $t$ is $\frac{d}{d t} [e^{t}] + \frac{d}{d t} [- 4]$ .

$\frac{d}{d t} [e^{t}] + \frac{d}{d t} [- 4]$

Step 1.1.3

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

$e^{t} + \frac{d}{d t} [- 4]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4$ with respect to $t$ is $0$ .

$e^{t} + 0$

Step 1.1.4.2

Add $e^{t}$ and $0$ .

$e^{t}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \cot (u) d u$

Step 2

The integral of $\cot (u)$ with respect to $u$ is $\ln (| \sin (u) |)$ .

$\ln (| \sin (u) |) + C$

Step 3

Replace all occurrences of $u$ with $e^{t} - 4$ .

$\ln (| \sin (e^{t} - 4) |) + C$