Question

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56.

${\int }_0^{\pi /2}\sec xdx$

Short answer

The answer is$\infty \left(Diverges\right).$

Step-by-step solution

Step 1. Given information.

The given integral is${\int }_0^{\pi /2}\sec xdx$.

Step 2. Value of the integral.

We know,

$\int secx=secx\tan x$

On simplifying,

$$\begin{gathered} {\int }_0^{\frac{\pi }{2}}\sec \left(x\right)dx=\left[\ln \right|\sec x+\tan x|{]}_0^{\frac{\pi }{2}} \\ =\left[\ln \left|\sec \left(\frac{\pi }{2}\right)+\tan \left(\frac{\pi }{2}\right)\right|-\ln \left|\sec \left(\frac{\pi }{2}\right)+\tan \left(\frac{\pi }{2}\right)\right|\right] \\ =\infty \\ (Diverges) \end{gathered}$$