Question

Find the value of $\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan xdx}$.
A. $\dfrac{\pi }{2}{{\log }_{e}}2$
B. $-\dfrac{\pi }{2}{{\log }_{e}}2$
C. $\pi {{\log }_{e}}2$
D. $0$

Answer 1

Hint: In this question, we have $\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan xdx}$. So, we will consider this equation as 1. Now we will use the identity $\int_{0}^{a}{f\left( x \right)dx=}\int_{0}^{a}{f\left( a-x \right)dx}$. After that, we will get equation 2.
Now we will add equations 1 and 2 and then we will use the identity $\tan \left( \dfrac{\pi }{2}-x \right)=\cot x$. At last, we will obtain the final result.



Formula Used:For the given problem, we will use the following formulas:
1) $\int_{0}^{a}{f\left( x \right)dx=}\int_{0}^{a}{f\left( a-x \right)dx}$
2) $\tan \left( \dfrac{\pi }{2}-x \right)=\cot x$
3) $\log \left( 1 \right)=0$



Complete step by step solution:We are given an integral as $\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan xdx}$.
Let us consider $I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan xdx}$ ….... (1)
First, we will use the identity $\int_{0}^{a}{f\left( x \right)dx=}\int_{0}^{a}{f\left( a-x \right)dx}$.
$I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan \left( \dfrac{\pi }{2}-x \right)}dx$
As we know that $\tan \left( \dfrac{\pi }{2}-x \right)=\cot x$
So, the integral becomes
$I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log \cot xdx}$ ….... (2)
Now, we are adding equations (1) and (2).
$\begin{align}
  & \Rightarrow 2I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log \tan xdx+\int\limits_{0}^{\dfrac{\pi }{2}}{\log \cot xdx}} \\
 & \Rightarrow 2I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log (\tan x\times \cot x)dx} \\
\end{align}$
Further, we will write $\cot x$ as the reciprocal of $\tan x$. So, we get
$\Rightarrow 2I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log \left( \tan x\times \dfrac{1}{\tan x} \right)}dx$
By canceling the same terms, we get
$\Rightarrow 2I=\int\limits_{0}^{\dfrac{\pi }{2}}{\log (1)dx}$
Remember that the value of $\log \left( 1 \right)=0$.
$\begin{align}
  & \Rightarrow 2I=\int\limits_{0}^{\dfrac{\pi }{2}}{(0)dx} \\
 & \Rightarrow 2I=0 \\
 & \therefore I=0 \\
\end{align}$



Option ‘D’ is correct



Note: As the functions change, it is not essential to integrate taking limits directly $0$ to $\dfrac{\pi }{2}$. We must avoid errors and confusion while dealing with sign changes that occurred during integration. We need to adopt the identities that make the problems easier. Students are advised to write the correct identity and then solve the problem. Don’t choose the identity that makes the question more complicated.