Calculus Examples

$\int_{\frac{π}{4}}^{\frac{3 π}{4}} - 8 \csc^{2} (x) d x$

Step 1

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$- 8 \int_{\frac{π}{4}}^{\frac{3 π}{4}} \csc^{2} (x) d x$

Step 2

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

$- 8 (- \cot (x)]_{\frac{π}{4}}^{\frac{3 π}{4}})$

Step 3

Simplify the answer.

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Step 3.1

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

$- 8 (- \cot (\frac{3 π}{4}) + \cot (\frac{π}{4}))$

Step 3.2

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$- 8 (- \cot (\frac{3 π}{4}) + 1)$

Step 3.3

Simplify.

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Step 3.3.1

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

$- 8 (- - \cot (\frac{π}{4}) + 1)$

Step 3.3.2

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$- 8 (- (- 1 \cdot 1) + 1)$

Step 3.3.3

Multiply $- 1$ by $1$ .

$- 8 (- - 1 + 1)$

Step 3.3.4

Multiply $- 1$ by $- 1$ .

$- 8 (1 + 1)$

Step 3.3.5

Add $1$ and $1$ .

$- 8 \cdot 2$

Step 3.3.6

Multiply $- 8$ by $2$ .

$- 16$