Question

Find the value of \[\int_0^\pi {x\log \sin x} dx\].
A. \[\dfrac{\pi }{2}\log \dfrac{1}{2}\]
B. \[ - \dfrac{{{\pi ^2}}}{2}\log 2\]
C. \[\pi \log \dfrac{1}{2}\]
D. \[{\pi ^2}\log \dfrac{1}{2}\]

Answer 1

Hint: In this question, we have \[\int_0^\pi {x\log \sin x} dx\]. 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 \[\int\limits_0^{2a} {f\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx} } \]. At last, we will obtain the final result.



Formula Used: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) \[\sin \left( {\pi - x} \right) = \sin x\]
3) \[\int\limits_0^{2a} {f\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx} } \]



Complete step by step solution:Let us consider \[I = \int_0^\pi {x\log \left( {\sin x} \right)} dx\] ….... (1)
First, we will use the identity \[\int_0^a {f\left( x \right)dx = } \int_0^a {f\left( {a - x} \right)dx} \].
\[I = \int_0^\pi {\left( {\pi - x} \right)\log \left( {\sin \left( {\pi - x} \right)} \right)} dx\]
As we know that \[\sin \left( {\pi - x} \right) = \sin x\].
So, we get
\[I = \int_0^\pi {\left( {\pi - x} \right)\log \left( {\sin x} \right)} dx\] ….... (2)
Now, we will add equations (1) and (2).
\[ \Rightarrow 2I = \int_0^\pi {\left( {x + \pi - x} \right)\log \left( {\sin x} \right)} dx\]
\[ \Rightarrow 2I = \pi \int_0^\pi {\log \left( {\sin x} \right)} dx\]
Further, we will use the identity \[\int\limits_0^{2a} {f\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx} } \] to get
\[ \Rightarrow 2I = \pi \cdot 2\int_0^{\dfrac{\pi }{2}} {\log \left( {\sin x} \right)} dx\]
\[ \Rightarrow 2I = 2\pi \int_0^{\dfrac{\pi }{2}} {\log \left( {\sin x} \right)} dx\]
We will further integrate 1 with respect to x.
\[ \Rightarrow I = \pi \left( { - \dfrac{\pi }{2}\log 2} \right)\]
\[ \Rightarrow I = - \dfrac{{{\pi ^2}}}{2}\log \left( 2 \right)\]



Option ‘C’ 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.