Question

Definite integrals: Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}{\sec }^{2}xdx$

Short answer

The solution is:$2$.

Step-by-step solution

Given 

The given integral is: ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}{\sec }^{2}xdx$.

To Find 

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

Calculation 

$$\begin{gathered} {\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}{\sec }^{2}xdx \\ ={\left[\tan \left(x\right)\right]}_{-\frac{\pi }{4}}^{\frac{\pi }{4}} \\ =\tan \left(\frac{\pi }{4}\right)-\tan \left(-\frac{\pi }{4}\right) \\ =\tan \left(\frac{\pi }{4}\right)+\tan \left(\frac{\pi }{4}\right) \\ =1+1 \\ =2 \end{gathered}$$

The sketch of the region is:

The shaded region is finite. So our answer seems reasonable.