Calculus Examples

$\int e^{4 \cot (x)} \cdot \csc^{2} (x) d x$

Step 1

Let $u = 4 \cot (x)$ . Then $d u = - 4 \csc^{2} (x) d x$ , so $- \frac{1}{4} d u = \csc^{2} (x) d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 4 \cot (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $4 \cot (x)$ .

$\frac{d}{d x} [4 \cot (x)]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 \cot (x)$ with respect to $x$ is $4 \frac{d}{d x} [\cot (x)]$ .

$4 \frac{d}{d x} [\cot (x)]$

Step 1.1.3

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$4 (- \csc^{2} (x))$

Step 1.1.4

Multiply $- 1$ by $4$ .

$- 4 \csc^{2} (x)$

Step 1.2

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

$\int e^{u} \frac{1}{- 4} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

$\int e^{u} (- \frac{1}{4}) d u$

Step 2.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$\int - \frac{e^{u}}{4} d u$

Step 3

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

$- \int \frac{e^{u}}{4} d u$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- (\frac{1}{4} \int e^{u} d u)$

Step 5

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- \frac{1}{4} (e^{u} + C)$

Step 6

Simplify.

$- \frac{1}{4} e^{u} + C$

Step 7

Replace all occurrences of $u$ with $4 \cot (x)$ .

$- \frac{1}{4} e^{4 \cot (x)} + C$