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Question Number 144052 by physicstutes last updated on 21/Jun/21

$$\mathrm{Evaluate}$$$$\int \frac{\sqrt{x}}{\sinh x}dx$$

Answered by mindispower last updated on 21/Jun/21

$$x>0$$$$=\int \frac{2e^{-x}\sqrt{x}dx}{1-e^{-2x}}$$$$=\sum _{k⩾0}\int 2e^{-(1+2k)x}\sqrt{x}dx$$$$=2\sum _{k⩾0}\frac{1}{{(1+2k)}^{\frac{3}{2}}}\int \sqrt{x}{e}^{-x}dx$$$$recal\mathrm{\Gamma }(a,x)incompletGammafunction$$$$\mathrm{\Gamma }(a,x)={\int }_x^{\mathrm{\infty }}{t}^{a-1}{e}^{-t}dt$$$$\int \sqrt{x}{e}^{-x}dx=\int x^{\frac{3}{2}-1}{e}^{-x}dx=-\mathrm{\Gamma }(\frac{3}{2},x)+c$$$$\sum _{m⩾0}\frac{1}{{(1+2k)}^{\frac{3}{2}}}=\zeta \left(\frac{3}{2}\right)-\frac{\zeta \left(\frac{3}{2}\right)}{2^{\frac{3}{2}}}$$$$weget,\int \frac{\sqrt{x}}{sh\left(x\right)}dx=-\frac{1}{\sqrt{2}}(2\sqrt{2}-1)\zeta \left(\frac{3}{2}\right)\mathrm{\Gamma }(\frac{3}{2},x)+c$$$$c\in \mathbb{R}$$

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