Calculus Examples

$\int_{\frac{1 π}{2}}^{2 π} \sec (x) \tan (x) d x$

Step 1

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

$\sec (x)]_{\frac{1 π}{2}}^{2 π}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\sec (x)$ at $2 π$ and at $\frac{1 π}{2}$ .

$(\sec (2 π)) - \sec (\frac{1 π}{2})$

Step 2.2

Multiply $π$ by $1$ .

$\sec (2 π) - \sec (\frac{π}{2})$

Step 3

Simplify each term.

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Step 3.1

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

$\sec (0) - \sec (\frac{π}{2})$

Step 3.2

The exact value of $\sec (0)$ is $1$ .

$1 - \sec (\frac{π}{2})$