Calculus Examples

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

Step 1

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 1.1.3

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

$3 \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

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

$\int \csc^{2} (u) \frac{1}{3} d u$

Step 2

Combine $\csc^{2} (u)$ and $\frac{1}{3}$ .

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \csc^{2} (u) d u$

Step 4

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

$\frac{1}{3} (- \cot (u) + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$\frac{1}{3} (- \cot (u)) + C$

Step 5.2

Combine $\frac{1}{3}$ and $\cot (u)$ .

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

Step 6

Replace all occurrences of $u$ with $3 x$ .

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

Step 7

Reorder terms.

$- \frac{1}{3} \cot (3 x) + C$