Definite integral of sin(lnx)/(lnx) from 0 to 1

Q&ACategory: IIT JEE AdvancedDefinite integral of sin(lnx)/(lnx) from 0 to 1
AY Sir Staff asked 1 year ago

Evaluate \int ^{1}_{0}\dfrac{sin\left( \ln x\right) }{\ln x}dx

1 Answers
AY Sir Staff answered 1 year ago
\begin{aligned}I\left( t\right) =\int ^{1}_{0}\dfrac{\sin \left( t\cdot \ln x\right) }{lnx}dx\Rightarrow I\left( 1\right) =\int ^{1}_{0}\dfrac{\sin \left( lnx\right) }{lnx}dx\\<br /> \Rightarrow I^{'}\left( t\right) =\int ^{1}_{0}\cos \left( t\cdot \ln x\right) dx=\int ^{1}_{0}Re\left( e^{itlnx}\right) dx\\<br /> \Rightarrow Re\left( \int ^{1}_{0}e^{lnx^{it}}dx\right) =Re\left( \int ^{1}_{0}x^{it}dx\right) =\dfrac{1}{1+t^{2}}\\<br /> \Rightarrow I'\left( t\right) =\dfrac{1}{1+t^{2}}\Rightarrow I\left( t\right) =\int ^{1}_{0}\dfrac{1}{1+t^{2}}dt\\<br /> \Rightarrow ( \tan ^{-1}\left( t\right) _{0}^{1}=\pi /4\end{aligned}