Calculus Examples

$\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 5 \csc (x) \cot (x) d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{\frac{7 π}{6}}^{\frac{11 π}{6}} \csc (x) \cot (x) d x$

Step 2

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

$5 (- \csc (x)]_{\frac{7 π}{6}}^{\frac{11 π}{6}})$

Step 3

Evaluate $- \csc (x)$ at $\frac{11 π}{6}$ and at $\frac{7 π}{6}$ .

$5 (- \csc (\frac{11 π}{6}) + \csc (\frac{7 π}{6}))$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$5 (- - \csc (\frac{π}{6}) + \csc (\frac{7 π}{6}))$

Step 4.1.2

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$5 (- (- 1 \cdot 2) + \csc (\frac{7 π}{6}))$

Step 4.1.3

Multiply $- (- 1 \cdot 2)$ .

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Step 4.1.3.1

Multiply $- 1$ by $2$ .

$5 (- - 2 + \csc (\frac{7 π}{6}))$

Step 4.1.3.2

Multiply $- 1$ by $- 2$ .

$5 (2 + \csc (\frac{7 π}{6}))$

Step 4.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the third quadrant.

$5 (2 - \csc (\frac{π}{6}))$

Step 4.1.5

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$5 (2 - 1 \cdot 2)$

Step 4.1.6

Multiply $- 1$ by $2$ .

$5 (2 - 2)$

Step 4.2

Subtract $2$ from $2$ .

$5 \cdot 0$

Step 4.3

Multiply $5$ by $0$ .

$0$