Calculus Examples

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

Step 1

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

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

Step 2

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$2 (\sec (x)]_{\frac{- π}{4}}^{\frac{3 π}{4}})$

Step 3

Substitute and simplify.

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Step 3.1

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

$2 ((\sec (\frac{3 π}{4})) - \sec (\frac{- π}{4}))$

Step 3.2

Move the negative in front of the fraction.

$2 (\sec (\frac{3 π}{4}) - \sec (- \frac{π}{4}))$

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 secant is negative in the second quadrant.

$2 (- \sec (\frac{π}{4}) - \sec (- \frac{π}{4}))$

Step 4.1.2

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

$2 (- \frac{2}{\sqrt{2}} - \sec (- \frac{π}{4}))$

Step 4.1.3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 (- \frac{2}{\sqrt{2}} - \sec (\frac{7 π}{4}))$

Step 4.1.4

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

$2 (- \frac{2}{\sqrt{2}} - \sec (\frac{π}{4}))$

Step 4.1.5

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

$2 (- \frac{2}{\sqrt{2}} - \frac{2}{\sqrt{2}})$

Step 4.2

Combine the numerators over the common denominator.

$2 \frac{- 2 - 2}{\sqrt{2}}$

Step 4.3

Subtract $2$ from $- 2$ .

$2 \frac{- 4}{\sqrt{2}}$

Step 4.4

Move the negative in front of the fraction.

$2 (- \frac{4}{\sqrt{2}})$

Step 4.5

Multiply $2 (- \frac{4}{\sqrt{2}})$ .

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Step 4.5.1

Multiply $- 1$ by $2$ .

$- 2 \frac{4}{\sqrt{2}}$

Step 4.5.2

Combine $- 2$ and $\frac{4}{\sqrt{2}}$ .

$\frac{- 2 \cdot 4}{\sqrt{2}}$

Step 4.5.3

Multiply $- 2$ by $4$ .

$\frac{- 8}{\sqrt{2}}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{8}{\sqrt{2}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{\sqrt{2}}$

Decimal Form:

$- 5.65685424 \dots$