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  4. ∫ limits ln x- ln 2 ln π (ex/1- cos ((2)3 ex)) d x is equal to

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Q. $\int\limits_{\ln x-\ln 2}^{\ln \pi} \frac{e^x}{1-\cos \left(\frac{2}{3} e^x\right)} d x$ is equal to

Integrals

Solution:

$\int\limits_{\ln\pi -\ln 2}^{\ln \pi} \frac{ e ^{ x }}{1-\cos \left(\frac{2}{3} e ^{ x }\right)} dx =\frac{3}{2} \int\limits_{\pi / 3}^{2 \pi / 3} \frac{ dt }{1-\cot }=\frac{3}{2} \int\limits_{\pi / 3}^{2 \pi / 3} \frac{ dt }{1-\left(1-2 \sin ^2 t \right)}$
$=\frac{3}{2.2} \int\limits_{\pi / 3}^{2 \pi / 3} \operatorname{cosec}^2 t dt =\frac{3}{4}[-\cot t ]_{\pi / 3}^{2 \pi / 3}$
$=-\frac{3}{4}\left[-\sqrt{3}-\frac{1}{\sqrt{3}}\right]=\frac{3}{4}\left(\frac{4}{\sqrt{3}}\right)=\sqrt{3}$