Calculus Examples

$\int_{\frac{π}{4}}^{\frac{3 π}{4}} \csc (x) \cot (x) d x$

Step 1

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

$- \csc (x)]_{\frac{π}{4}}^{\frac{3 π}{4}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $- \csc (x)$ at $\frac{3 π}{4}$ and at $\frac{π}{4}$ .

$- \csc (\frac{3 π}{4}) + \csc (\frac{π}{4})$

Step 2.2

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$- \csc (\frac{3 π}{4}) + \sqrt{2}$

Step 2.3

Simplify.

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Step 2.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \csc (\frac{π}{4}) + \sqrt{2}$

Step 2.3.2

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$- \sqrt{2} + \sqrt{2}$

Step 2.3.3

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$0$