Question

The value of \[\;\int\limits_0^\infty {\frac{1}{{\left( {{a^2}\; + {\text{ }}{x^2}} \right)}}dx} \;\]is,
(1) \[\frac{\pi }{2}\]
(2) \[\frac{\pi }{{2a}}\]
(3) \[\frac{\pi }{a}\]
(4) \[\frac{1}{{2a}}\]

Answer 1

Hint: The question is from the chapter, named definite Integration. This is a formula-based question. Directly, apply the formula, and after that put the limit as given into the question.

Formula Used: 1) \[\begin{array}{*{20}{c}}
  {\int {\frac{1}{{({a^2} + {x^2})}}} }& = &{\frac{1}{a}{{\tan }^{ - 1}}\frac{x}{a}}
\end{array}\]
2) \[\begin{array}{*{20}{c}}
  {{{\tan }^{ - 1}}\left( {\tan \theta } \right)}& = &\theta
\end{array}\]

Complete step by step Solution:
Integral is the area under a curve. In other words, Integral is the summation of all the small areas of a curve. Basically, integral helps to find the area of a non-uniform geometry or a body. Similarly, a definite integral can also be defined as the area under a curve between two fixed points.
Now, given that
\[ \Rightarrow \;\int\limits_0^\infty {\frac{1}{{\left( {{a^2}\; + {\text{ }}{x^2}} \right)}}dx} \;\]
Therefore, we know that the integral of \[\frac{1}{{\left( {{a^2}\; + {\text{ }}{x^2}} \right)}}\]is \[\frac{1}{a}{\tan ^{ - 1}}\frac{x}{a}\]. Therefore, we will get
\[ \Rightarrow \frac{1}{a}\left[ {{{\tan }^{ - 1}}\frac{x}{a}} \right]_0^\infty \]
When we solve the definite integral, we will have to put the limit after getting the integral of an expression. There are two limits. One is the lower limit and the other is called the upper limit.
According to the given question, 0 is the lower limit and \[\infty \]is the upper limit. So, we will get,
\[ \Rightarrow \frac{1}{a}\left[ {{{\tan }^{ - 1}}\frac{\infty }{a} - {{\tan }^{ - 1}}\frac{0}{a}} \right]\]
And
\[ \Rightarrow \frac{1}{a}\left[ {{{\tan }^{ - 1}}\infty - {{\tan }^{ - 1}}0} \right]\]…… (a)
Form the trigonometric table,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \tan \frac{\pi }{2}}& = &\infty
\end{array}\]and \[\begin{array}{*{20}{c}}
  {\tan 0}& = &0
\end{array}\]
Now, from equation (a). we can write,
\[ \Rightarrow \frac{1}{a}\left[ {{{\tan }^{ - 1}}\left( {\tan \frac{\pi }{2}} \right) - {{\tan }^{ - 1}}\left( {\tan 0} \right)} \right]\]
There will be some trigonometric properties that will be used. So,
\[\begin{array}{*{20}{c}}
  { \Rightarrow {{\tan }^{ - 1}}\left( {\tan \theta } \right)}& = &\theta
\end{array}\]
Therefore, we can write.
\[ \Rightarrow \frac{1}{a}\left[ {\frac{\pi }{2} - 0} \right]\]
\[ \Rightarrow \frac{\pi }{{2a}}\]
Now the final answer is \[\frac{\pi }{{2a}}\].

Hence, the correct option is 2.

Note:When we solve the indefinite integral, there is a constant that comes out while solving the definite integral, there is no constant that comes out because the definite integral finds the area under the curve between the two fixed points.