Calculus Examples

$\int \cot^{2} (7 x) d x$

Step 1

Let $u = 7 x$ . Then $d u = 7 d x$ , so $\frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 7 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 1.1.2

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

$7 \frac{d}{d x} [x]$

Step 1.1.3

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

$7 \cdot 1$

Step 1.1.4

Multiply $7$ by $1$ .

$7$

Step 1.2

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

$\int \cot^{2} (u) \frac{1}{7} d u$

Step 2

Combine $\cot^{2} (u)$ and $\frac{1}{7}$ .

$\int \frac{\cot^{2} (u)}{7} d u$

Step 3

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int \cot^{2} (u) d u$

Step 4

Using the Pythagorean Identity, rewrite $\cot^{2} (u)$ as $- 1 + \csc^{2} (u)$ .

$\frac{1}{7} \int - 1 + \csc^{2} (u) d u$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{7} (\int - 1 d u + \int \csc^{2} (u) d u)$

Step 6

Apply the constant rule.

$\frac{1}{7} (- u + C + \int \csc^{2} (u) d u)$

Step 7

Since the derivative of $- \cot (u)$ is $\csc^{2} (u)$ , the integral of $\csc^{2} (u)$ is $- \cot (u)$ .

$\frac{1}{7} (- u + C - \cot (u) + C)$

Step 8

Simplify.

$\frac{1}{7} (- u - \cot (u)) + C$

Step 9

Replace all occurrences of $u$ with $7 x$ .

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

Step 10

Simplify.

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Step 10.1

Multiply $7$ by $- 1$ .

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Cancel the common factor of $7$ .

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Step 10.3.1

Factor $7$ out of $- 7 x$ .

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

Step 10.3.2

Cancel the common factor.

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

Step 10.3.3

Rewrite the expression.

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

Step 10.4

Combine $\frac{1}{7}$ and $\cot (7 x)$ .

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

Step 11

Reorder terms.

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