Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x$.[2] Show that \(\dfrac{\m

Question

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x$.[2] Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\cos3x)=-3\tan3x$.[2] Show that $\cot6x=\dfrac12(\cot3x-\tan3x)$.[2] Hence evaluate the exact value of $\displaystyle\int_{\pi/18}^{\pi/12}\cot6x\,\mathrm{d}x$.[4]

Accepted Answer

(iv) $\dfrac16\ln\left(\dfrac2{\sqrt3}\right)$

Cotangent integral from logarithmic derivatives

Cotangent integral from logarithmic derivatives

Video Solution:

Solution:

Solution locked