Chapter 7, Problem 1RE

To determine

To evaluate the integral.

Answer to Problem 1RE

The required solution is , ${\displaystyle {\int }_0^{\pi /3}{\sec }^2\theta d\theta }=\sqrt{3}$ .

Explanation of Solution

Given information: The equation is, ${\displaystyle \underset{0}{\overset{\pi /3}{\int }}{\sec }^2\theta }d\theta$

Concept used: Definite integrals,

The definite integral of $f\left(x\right)$ over $\left[a,b\right]$ , denoted by ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx,}$ is given by,

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx,}={\left[F\left(\text{b}\right)\right]}_a^b=F\left(b\right)-F\left(a\right)$

Calculation : The required solution is obtained as,

  $\begin{array}{l}{\displaystyle {\int }_0^{\pi /3}{\sec }^2\theta d\theta }\\ ={\left[\tan \theta \right]}_0^{\pi /3}\\ =\tan \left(\frac{\pi }{3}\right)-\tan \left(0\right)\\ =\sqrt{3}-0\\ =\sqrt{3}\end{array}$

Hence, the required solution is ${\displaystyle {\int }_0^{\pi /3}{\sec }^2\theta d\theta }=\sqrt{3}$ .