Calculus Examples

$\int 9 \csc^{2} (6 x - 2) d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int \csc^{2} (6 x - 2) d x$

Step 2

Let $u = 6 x - 2$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 6 x - 2$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $6 x - 2$ .

$\frac{d}{d x} [6 x - 2]$

Step 2.1.2

By the Sum Rule, the derivative of $6 x - 2$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 2]$

Step 2.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [- 2]$

Step 2.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 2]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$6 + 0$

Step 2.1.4.2

Add $6$ and $0$ .

$6$

Step 2.2

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

$9 \int \csc^{2} (u) \frac{1}{6} d u$

Step 3

Combine $\csc^{2} (u)$ and $\frac{1}{6}$ .

$9 \int \frac{\csc^{2} (u)}{6} d u$

Step 4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$9 (\frac{1}{6} \int \csc^{2} (u) d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{6}$ and $9$ .

$\frac{9}{6} \int \csc^{2} (u) d u$

Step 5.2

Cancel the common factor of $9$ and $6$ .

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Step 5.2.1

Factor $3$ out of $9$ .

$\frac{3 (3)}{6} \int \csc^{2} (u) d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 3}{3 \cdot 2} \int \csc^{2} (u) d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 \cdot 2} \int \csc^{2} (u) d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{3}{2} \int \csc^{2} (u) d u$

Step 6

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

$\frac{3}{2} (- \cot (u) + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{3}{2} (- \cot (u)) + C$

Step 7.2

Combine $\frac{3}{2}$ and $\cot (u)$ .

$- \frac{3 \cot (u)}{2} + C$

Step 8

Replace all occurrences of $u$ with $6 x - 2$ .

$- \frac{3 \cot (6 x - 2)}{2} + C$

Step 9

Reorder terms.

$- \frac{3}{2} \cot (6 x - 2) + C$