Calculus Examples

$\int \csc (- 7 x) \cdot \cot (- 7 x) d x$

Step 1

Let $u_{2} = \csc (- 7 x)$ . Then $d u_{2} = 7 \cot (- 7 x) \csc (- 7 x) d x$ , so $\frac{1}{7} d u_{2} = \cot (- 7 x) \csc (- 7 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 1.1

Let $u_{2} = \csc (- 7 x)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\csc (- 7 x)$ .

$\frac{d}{d x} [\csc (- 7 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = - 7 x$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- 7 x$ .

$\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [- 7 x]$

Step 1.1.2.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [- 7 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $- 7 x$ .

$- \csc (- 7 x) \cot (- 7 x) \frac{d}{d x} [- 7 x]$

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- \csc (- 7 x) \cot (- 7 x) (- 7 \frac{d}{d x} [x])$

Step 1.1.3.2

Multiply $- 7$ by $- 1$ .

$7 \csc (- 7 x) \cot (- 7 x) \frac{d}{d x} [x]$

Step 1.1.3.3

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

$7 \csc (- 7 x) \cot (- 7 x) \cdot 1$

Step 1.1.3.4

Simplify the expression.

Tap for more steps...

Step 1.1.3.4.1

Multiply $7$ by $1$ .

$7 \csc (- 7 x) \cot (- 7 x)$

Step 1.1.3.4.2

Reorder the factors of $7 \csc (- 7 x) \cot (- 7 x)$ .

$7 \cot (- 7 x) \csc (- 7 x)$

Step 1.2

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

$\int \frac{1}{7} d u_{2}$

Step 2

Apply the constant rule.

$\frac{1}{7} u_{2} + C$

Step 3

Replace all occurrences of $u_{2}$ with $\csc (- 7 x)$ .

$\frac{1}{7} \csc (- 7 x) + C$