Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$4 (2 - \sec (- \frac{π}{3}))$

Step 3.3

Simplify.

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Step 3.3.1

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

$4 (2 - \sec (\frac{5 π}{3}))$

Step 3.3.2

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

$4 (2 - \sec (\frac{π}{3}))$

Step 3.3.3

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$4 (2 - 1 \cdot 2)$

Step 3.3.4

Multiply $- 1$ by $2$ .

$4 (2 - 2)$

Step 3.3.5

Subtract $2$ from $2$ .

$4 \cdot 0$

Step 3.3.6

Multiply $4$ by $0$ .

$0$