Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(\csc (x) - \tan (x))}^{2}$ as $(\csc (x) - \tan (x)) (\csc (x) - \tan (x))$ .

$\int (\csc (x) - \tan (x)) (\csc (x) - \tan (x)) d x$

Step 1.2

Expand $(\csc (x) - \tan (x)) (\csc (x) - \tan (x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\int \csc (x) (\csc (x) - \tan (x)) - \tan (x) (\csc (x) - \tan (x)) d x$

Step 1.2.2

Apply the distributive property.

$\int \csc (x) \csc (x) + \csc (x) (- \tan (x)) - \tan (x) (\csc (x) - \tan (x)) d x$

Step 1.2.3

Apply the distributive property.

$\int \csc (x) \csc (x) + \csc (x) (- \tan (x)) - \tan (x) \csc (x) - \tan (x) (- \tan (x)) d x$

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 $\csc (x) \csc (x)$ .

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

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 $\csc (x)$ and $- \tan (x)$ .

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

Step 1.3.1.2.2

Add parentheses.

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

Step 1.3.1.2.3

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

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

Step 1.3.1.2.4

Cancel the common factors.

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

Step 1.3.1.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

Add parentheses.

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

Step 1.3.1.4.2

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

$\int \csc^{2} (x) - \sec (x) - (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin (x)}) - \tan (x) (- \tan (x)) d x$

Step 1.3.1.4.3

Cancel the common factors.

$\int \csc^{2} (x) - \sec (x) - \frac{1}{\cos (x)} - \tan (x) (- \tan (x)) d x$

Step 1.3.1.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.3.1.6

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

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.6.2

Multiply $\tan (x)$ by $1$ .

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

Step 1.3.1.6.3

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

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

Step 1.3.1.6.4

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

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

Step 1.3.1.6.5

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

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

Step 1.3.1.6.6

Add $1$ and $1$ .

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

Step 1.3.2

Subtract $\sec (x)$ from $- \sec (x)$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 5

The integral of $\sec (x)$ with respect to $x$ is $\ln (| \sec (x) + \tan (x) |)$ .

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

Step 6

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

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

$- \cot (x) + C - 2 (\ln (| \sec (x) + \tan (x) |) + C) - x + C + \tan (x) + C$

Step 10

Simplify.

$- \cot (x) - 2 \ln (| \sec (x) + \tan (x) |) - x + \tan (x) + C$