Calculus Examples

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

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

$\int \frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + \sin^{2} (x) + \cos^{2} (x)} d x$ is an antiderivative of $\frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + \sin^{2} (x) + \cos^{2} (x)}$ , so by definition $\frac{d}{d x} [\int \frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + \sin^{2} (x) + \cos^{2} (x)} d x]$ is $\frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + \sin^{2} (x) + \cos^{2} (x)}$ .

$\frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + \sin^{2} (x) + \cos^{2} (x)}$

Step 3

Simplify.

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Step 3.1

Apply pythagorean identity.

$\frac{\sec (x) + \csc (x) + \cot (x)}{\tan^{2} (x) + 1}$

Step 3.2

Apply pythagorean identity.

$\frac{\sec (x) + \csc (x) + \cot (x)}{\sec^{2} (x)}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

$\frac{\frac{1}{\cos (x)} + \csc (x) + \cot (x)}{\sec^{2} (x)}$

Step 3.3.2

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

$\frac{\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \cot (x)}{\sec^{2} (x)}$

Step 3.3.3

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

$\frac{\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\sec^{2} (x)}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

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

$\frac{\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{{(\frac{1}{\cos (x)})}^{2}}$

Step 3.4.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\frac{1^{2}}{\cos^{2} (x)}}$

Step 3.4.3

One to any power is one.

$\frac{\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\frac{1}{\cos^{2} (x)}}$

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}) \cos^{2} (x)$

Step 3.6

Apply the distributive property.

$\frac{1}{\cos (x)} \cos^{2} (x) + \frac{1}{\sin (x)} \cos^{2} (x) + \frac{\cos (x)}{\sin (x)} \cos^{2} (x)$

Step 3.7

Simplify.

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Step 3.7.1

Cancel the common factor of $\cos (x)$ .

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Step 3.7.1.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{1}{\cos (x)} (\cos (x) \cos (x)) + \frac{1}{\sin (x)} \cos^{2} (x) + \frac{\cos (x)}{\sin (x)} \cos^{2} (x)$

Step 3.7.1.2

Cancel the common factor.

$\frac{1}{\cos (x)} (\cos (x) \cos (x)) + \frac{1}{\sin (x)} \cos^{2} (x) + \frac{\cos (x)}{\sin (x)} \cos^{2} (x)$

Step 3.7.1.3

Rewrite the expression.

$\cos (x) + \frac{1}{\sin (x)} \cos^{2} (x) + \frac{\cos (x)}{\sin (x)} \cos^{2} (x)$

Step 3.7.2

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

$\cos (x) + \frac{\cos^{2} (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cos^{2} (x)$

Step 3.7.3

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

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Step 3.7.3.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $\cos^{2} (x)$ .

$\cos (x) + \frac{\cos^{2} (x)}{\sin (x)} + \frac{\cos (x) \cos^{2} (x)}{\sin (x)}$

Step 3.7.3.2

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 3.7.3.2.1

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

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Step 3.7.3.2.1.1

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

$\cos (x) + \frac{\cos^{2} (x)}{\sin (x)} + \frac{\cos^{1} (x) \cos^{2} (x)}{\sin (x)}$

Step 3.7.3.2.1.2

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

$\cos (x) + \frac{\cos^{2} (x)}{\sin (x)} + \frac{{\cos (x)}^{1 + 2}}{\sin (x)}$

Step 3.7.3.2.2

Add $1$ and $2$ .

$\cos (x) + \frac{\cos^{2} (x)}{\sin (x)} + \frac{\cos^{3} (x)}{\sin (x)}$

Step 3.8

Simplify each term.

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Step 3.8.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\cos (x) + \frac{\cos (x) \cos (x)}{\sin (x)} + \frac{\cos^{3} (x)}{\sin (x)}$

Step 3.8.2

Separate fractions.

$\cos (x) + \frac{\cos (x)}{1} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cos^{3} (x)}{\sin (x)}$

Step 3.8.3

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

$\cos (x) + \frac{\cos (x)}{1} \cot (x) + \frac{\cos^{3} (x)}{\sin (x)}$

Step 3.8.4

Divide $\cos (x)$ by $1$ .

$\cos (x) + \cos (x) \cot (x) + \frac{\cos^{3} (x)}{\sin (x)}$

Step 3.8.5

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$\cos (x) + \cos (x) \cot (x) + \frac{\cos (x) \cos^{2} (x)}{\sin (x)}$

Step 3.8.6

Separate fractions.

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

Step 3.8.7

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

$\cos (x) + \cos (x) \cot (x) + \frac{\cos^{2} (x)}{1} \cot (x)$

Step 3.8.8

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

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