Question

If ${\Phi }^{\prime }\left(x\right)=\frac{log\left(sinx\right)}{x},x\ne n\pi ,nϵIand{\int }_1^3\frac{3log\left(sinx\right)}{x}dx=\Phi \left(k\right)-\Phi \left(1\right)$ then the value of k is

• A. 27
• B. 18
• C. 9
• D. None of these

Answer 1

The correct option is A 27

${\int }_1^3\frac{3log\left(sin{x}^3\right)}{x}dx={\int }_1^{27}\frac{log\left(sinz\right)}{z}dz$
$[putting{x}^3=z\Rightarrow 3dx=\frac{dz}{x^2}]$
$={\int }_1^{27}\Phi \left(z\right)dz=\left[\Phi \right(z){]}_1^{27}=\Phi (27)-\Phi (1)$
$\therefore k=27$