Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

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

Step 2

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

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

Apply the reduction formula.

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

Step 5

The integral of $\cot (u)$ with respect to $u$ is $\ln (| \sin (u) |)$ .

$\frac{1}{2} (- \frac{\cot^{2} (u)}{2} - (\ln (| \sin (u) |) + C))$

Step 6

Rewrite $\frac{1}{2} (- \frac{\cot^{2} (u)}{2} - (\ln (| \sin (u) |) + C))$ as $\frac{1}{2} (- \frac{1}{2} \cot^{2} (u) - \ln (| \sin (u) |)) + C$ .

$\frac{1}{2} (- \frac{1}{2} \cot^{2} (u) - \ln (| \sin (u) |)) + C$

Step 7

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

$\frac{1}{2} (- \frac{1}{2} \cot^{2} (2 x) - \ln (| \sin (2 x) |)) + C$

Step 8

Simplify.

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Step 8.1

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

$\frac{1}{2} (- \frac{\cot^{2} (2 x)}{2} - \ln (| \sin (2 x) |)) + C$

Step 8.2

Apply the distributive property.

$\frac{1}{2} (- \frac{\cot^{2} (2 x)}{2}) + \frac{1}{2} (- \ln (| \sin (2 x) |)) + C$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Rewrite $\cot (2 x)$ in terms of sines and cosines.

$\frac{1}{2} (- \frac{{(\frac{\cos (2 x)}{\sin (2 x)})}^{2}}{2}) + \frac{1}{2} (- \ln (| \sin (2 x) |)) + C$

Step 8.3.2

Apply the product rule to $\frac{\cos (2 x)}{\sin (2 x)}$ .

$\frac{1}{2} (- \frac{\frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}}{2}) + \frac{1}{2} (- \ln (| \sin (2 x) |)) + C$

Step 8.4

Combine $\frac{1}{2}$ and $\ln (| \sin (2 x) |)$ .

$\frac{1}{2} (- \frac{\frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}}{2}) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5

Simplify each term.

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Step 8.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (- (\frac{\cos^{2} (2 x)}{\sin^{2} (2 x)} \cdot \frac{1}{2})) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5.2

Combine.

$\frac{1}{2} (- \frac{\cos^{2} (2 x) \cdot 1}{\sin^{2} (2 x) \cdot 2}) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5.3

Multiply $\cos^{2} (2 x)$ by $1$ .

$\frac{1}{2} (- \frac{\cos^{2} (2 x)}{\sin^{2} (2 x) \cdot 2}) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5.4

Move $2$ to the left of $\sin^{2} (2 x)$ .

$\frac{1}{2} (- \frac{\cos^{2} (2 x)}{2 \cdot \sin^{2} (2 x)}) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5.5

Multiply $\frac{1}{2} (- \frac{\cos^{2} (2 x)}{2 \sin^{2} (2 x)})$ .

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Step 8.5.5.1

Multiply $\frac{1}{2}$ by $\frac{\cos^{2} (2 x)}{2 \sin^{2} (2 x)}$ .

$- \frac{\cos^{2} (2 x)}{2 (2 \sin^{2} (2 x))} - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.5.5.2

Multiply $2$ by $2$ .

$- \frac{\cos^{2} (2 x)}{4 \sin^{2} (2 x)} - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6

Simplify each term.

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Step 8.6.1

Multiply by $1$ .

$- \frac{\cos^{2} (2 x) \cdot 1}{4 \sin^{2} (2 x)} - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6.2

Multiply by $1$ .

$- \frac{\cos^{2} (2 x) \cdot 1}{4 (\sin^{2} (2 x) \cdot 1)} - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6.3

Separate fractions.

$- (\frac{1}{4 \cdot (1)} \cdot \frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6.4

Convert from $\frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}$ to $\cot^{2} (2 x)$ .

$- (\frac{1}{4 \cdot (1)} \cot^{2} (2 x)) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6.5

Multiply $4$ by $1$ .

$- (\frac{1}{4} \cot^{2} (2 x)) - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 8.6.6

Combine $\frac{1}{4}$ and $\cot^{2} (2 x)$ .

$- \frac{\cot^{2} (2 x)}{4} - \frac{\ln (| \sin (2 x) |)}{2} + C$

Step 9

Reorder terms.

$- \frac{1}{4} \cot^{2} (2 x) - \frac{1}{2} \ln (| \sin (2 x) |) + C$