Calculus Examples

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

Step 1

Write ${(\tan (x) + \cot (x))}^{2}$ as a function.

$f (x) = {(\tan (x) + \cot (x))}^{2}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

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

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

Step 4.3.1.1.2

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

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

Step 4.3.1.1.3

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

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

Step 4.3.1.1.4

Add $1$ and $1$ .

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

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

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

Step 4.3.1.2.3

Cancel the common factors.

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

Step 4.3.1.3

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

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Step 4.3.1.3.1

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

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

Step 4.3.1.3.2

Cancel the common factors.

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

Step 4.3.1.4

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

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Step 4.3.1.4.1

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

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

Step 4.3.1.4.2

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

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

Step 4.3.1.4.3

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

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

Step 4.3.1.4.4

Add $1$ and $1$ .

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

Step 4.3.2

Add $1$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

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

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

Apply the constant rule.

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

Step 14

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

$- x + C + \tan (x) + C + 2 x + C - x + C - \cot (x) + C$

Step 15

Simplify.

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Step 15.1

Simplify.

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Step 15.1.1

Add $- x$ and $2 x$ .

$C + \tan (x) + C + x + C - x + C - \cot (x) + C$

Step 15.1.2

Subtract $x$ from $x$ .

$C + \tan (x) + C + C + 0 + C - \cot (x) + C$

Step 15.1.3

Add $C + \tan (x) + C + C$ and $0$ .

$C + \tan (x) + C + C + C - \cot (x) + C$

Step 15.2

Simplify.

$\tan (x) - \cot (x) + C$

Step 16

The answer is the antiderivative of the function $f (x) = {(\tan (x) + \cot (x))}^{2}$ .

$F (x) =$ $\tan (x) - \cot (x) + C$