Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(\tan (2 x) + \cot (2 x))}^{2}$ as $(\tan (2 x) + \cot (2 x)) (\tan (2 x) + \cot (2 x))$ .

$\int (\tan (2 x) + \cot (2 x)) (\tan (2 x) + \cot (2 x)) d x$

Step 1.2

Expand $(\tan (2 x) + \cot (2 x)) (\tan (2 x) + \cot (2 x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\int \tan (2 x) (\tan (2 x) + \cot (2 x)) + \cot (2 x) (\tan (2 x) + \cot (2 x)) d x$

Step 1.2.2

Apply the distributive property.

$\int \tan (2 x) \tan (2 x) + \tan (2 x) \cot (2 x) + \cot (2 x) (\tan (2 x) + \cot (2 x)) d x$

Step 1.2.3

Apply the distributive property.

$\int \tan (2 x) \tan (2 x) + \tan (2 x) \cot (2 x) + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $\tan (2 x) \tan (2 x)$ .

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Step 1.3.1.1.1

Raise $\tan (2 x)$ to the power of $1$ .

$\int \tan^{1} (2 x) \tan (2 x) + \tan (2 x) \cot (2 x) + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.1.2

Raise $\tan (2 x)$ to the power of $1$ .

$\int \tan^{1} (2 x) \tan^{1} (2 x) + \tan (2 x) \cot (2 x) + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int {\tan (2 x)}^{1 + 1} + \tan (2 x) \cot (2 x) + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1.2.1

Reorder $\tan (2 x)$ and $\cot (2 x)$ .

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

Step 1.3.1.2.2

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

$\int \tan^{2} (2 x) + \frac{\cos (2 x)}{\sin (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.2.3

Cancel the common factors.

$\int \tan^{2} (2 x) + 1 + \cot (2 x) \tan (2 x) + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1.3.1

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

$\int \tan^{2} (2 x) + 1 + \frac{\cos (2 x)}{\sin (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.3.2

Cancel the common factors.

$\int \tan^{2} (2 x) + 1 + 1 + \cot (2 x) \cot (2 x) d x$

Step 1.3.1.4

Multiply $\cot (2 x) \cot (2 x)$ .

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Step 1.3.1.4.1

Raise $\cot (2 x)$ to the power of $1$ .

$\int \tan^{2} (2 x) + 1 + 1 + \cot^{1} (2 x) \cot (2 x) d x$

Step 1.3.1.4.2

Raise $\cot (2 x)$ to the power of $1$ .

$\int \tan^{2} (2 x) + 1 + 1 + \cot^{1} (2 x) \cot^{1} (2 x) d x$

Step 1.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \tan^{2} (2 x) + 1 + 1 + {\cot (2 x)}^{1 + 1} d x$

Step 1.3.1.4.4

Add $1$ and $1$ .

$\int \tan^{2} (2 x) + 1 + 1 + \cot^{2} (2 x) d x$

Step 1.3.2

Add $1$ and $1$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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Step 3.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $2 x$ .

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

Step 3.1.2

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 3.1.3

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 3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \tan^{2} (u_{1}) \frac{1}{2} d u_{1} + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 4

Combine $\tan^{2} (u_{1})$ and $\frac{1}{2}$ .

$\int \frac{\tan^{2} (u_{1})}{2} d u_{1} + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 5

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

$\frac{1}{2} \int \tan^{2} (u_{1}) d u_{1} + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 6

Using the Pythagorean Identity, rewrite $\tan^{2} (u_{1})$ as $- 1 + \sec^{2} (u_{1})$ .

$\frac{1}{2} \int - 1 + \sec^{2} (u_{1}) d u_{1} + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{2} (\int - 1 d u_{1} + \int \sec^{2} (u_{1}) d u_{1}) + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 8

Apply the constant rule.

$\frac{1}{2} (- u_{1} + C + \int \sec^{2} (u_{1}) d u_{1}) + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 9

Since the derivative of $\tan (u_{1})$ is $\sec^{2} (u_{1})$ , the integral of $\sec^{2} (u_{1})$ is $\tan (u_{1})$ .

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + \int 2 d x + \int \cot^{2} (2 x) d x$

Step 10

Apply the constant rule.

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \int \cot^{2} (2 x) d x$

Step 11

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

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Step 11.1

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

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Step 11.1.1

Differentiate $2 x$ .

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

Step 11.1.2

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 11.1.3

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 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \int \cot^{2} (u_{2}) \frac{1}{2} d u_{2}$

Step 12

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

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \int \frac{\cot^{2} (u_{2})}{2} d u_{2}$

Step 13

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

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \frac{1}{2} \int \cot^{2} (u_{2}) d u_{2}$

Step 14

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

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \frac{1}{2} \int - 1 + \csc^{2} (u_{2}) d u_{2}$

Step 15

Split the single integral into multiple integrals.

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \frac{1}{2} (\int - 1 d u_{2} + \int \csc^{2} (u_{2}) d u_{2})$

Step 16

Apply the constant rule.

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \frac{1}{2} (- u_{2} + C + \int \csc^{2} (u_{2}) d u_{2})$

Step 17

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

$\frac{1}{2} (- u_{1} + C + \tan (u_{1}) + C) + 2 x + C + \frac{1}{2} (- u_{2} + C - \cot (u_{2}) + C)$

Step 18

Simplify.

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Step 18.1

Simplify.

$- \frac{u_{1}}{2} + \frac{\tan (u_{1})}{2} + 2 x + \frac{1}{2} (- u_{2} - \cot (u_{2})) + C$

Step 18.2

Simplify.

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Step 18.2.1

To write $\frac{1}{2} (- u_{2} - \cot (u_{2}))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1})}{2} + \frac{1}{2} (- u_{2} - \cot (u_{2})) \cdot \frac{2}{2} + C$

Step 18.2.2

Combine $\frac{1}{2} (- u_{2} - \cot (u_{2}))$ and $\frac{2}{2}$ .

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1})}{2} + \frac{\frac{1}{2} (- u_{2} - \cot (u_{2})) \cdot 2}{2} + C$

Step 18.2.3

Combine the numerators over the common denominator.

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1}) + \frac{1}{2} (- u_{2} - \cot (u_{2})) \cdot 2}{2} + C$

Step 18.2.4

Combine $2$ and $\frac{1}{2}$ .

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1}) + \frac{2}{2} (- u_{2} - \cot (u_{2}))}{2} + C$

Step 18.2.5

Cancel the common factor of $2$ .

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Step 18.2.5.1

Cancel the common factor.

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1}) + \frac{2}{2} (- u_{2} - \cot (u_{2}))}{2} + C$

Step 18.2.5.2

Rewrite the expression.

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1}) + 1 (- u_{2} - \cot (u_{2}))}{2} + C$

Step 18.2.6

Multiply $- u_{2} - \cot (u_{2})$ by $1$ .

$- \frac{u_{1}}{2} + 2 x + \frac{\tan (u_{1}) - u_{2} - \cot (u_{2})}{2} + C$

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

Replace all occurrences of $u_{1}$ with $2 x$ .

$- \frac{2 x}{2} + 2 x + \frac{\tan (2 x) - u_{2} - \cot (u_{2})}{2} + C$

Step 19.2

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 20

Simplify.

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Step 20.1

Reduce the expression $\frac{2 x}{2}$ by cancelling the common factors.

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Step 20.1.1

Cancel the common factor.

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

Step 20.1.2

Rewrite the expression.

$- \frac{x}{1} + 2 x + \frac{\tan (2 x) - (2 x) - \cot (2 x)}{2} + C$

Step 20.2

Divide $x$ by $1$ .

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

Step 20.3

Add $- x$ and $2 x$ .

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

Step 20.4

Multiply $2$ by $- 1$ .

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

Step 21

Reorder terms.

$x + \frac{1}{2} (\tan (2 x) - 2 x - \cot (2 x)) + C$