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Question Number 136060 by mnjuly1970 last updated on 18/Mar/21

$$...advancedcalculus....$$$$evaluate::$$$$\mathit{\varphi }=\mathrm{Im}({\int }_0^{\frac{\pi }{2}}{li}_2\left(sin\left(x\right)\right)+{li}_2\left(csc\left(x\right)\right)dx)$$$$$$

Answered by mindispower last updated on 18/Mar/21

$$csc\left(x\right)=\frac{1}{sin\left(x\right)}$$$$wehave$$$${li}_2\left(z\right)+{li}_2\left(\frac{1}{z}\right)=-\frac{{\pi }^2}{6}-\frac{{ln}^2(-z)}{2}$$$$\iff \varphi =Im{\int }_0^{\frac{\pi }{2}}(-\frac{{\mathit{\pi }}^2}{6}-\frac{{ln}^2(-sin\left(x\right))}{2})dx$$$$=-\frac{Im}{2}{\int }_0^{\frac{\pi }{2}}{(ln\left(sin\left(x\right)\right)+i\pi )}^{2}dx$$$$=-\frac{Im}{2}{\int }_0^{\frac{\pi }{2}}\left(2i\pi ln\left(sin\left(x\right)\right)\right)dx$$$$=-\pi {\int }_0^{\frac{\pi }{2}}ln\left(sin\left(x\right)\right)dx=\frac{{\pi }^{2}log\left(2\right)}{2}$$

Commented by mnjuly1970 last updated on 18/Mar/21

$$thanksalotsirpower...$$$$$$

Commented by mindispower last updated on 19/Mar/21

$$pleasursir$$$$$$

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