タイトル未設定

$$\begin{array}{rlr}{\int }_0^1\frac{x^{\alpha }-1}{\ln x}dx& =-{\int }_0^{\mathrm{\infty }}\frac{e^{-\alpha t}-1}{t{e}^t}dt& (x\mapsto e^{-t})\\ & =-{\int }_0^{\mathrm{\infty }}\frac{1}{t{e}^t}{\int }_0^{-\alpha }\frac{\partial }{\partial s}(e^{st})dsdt\\ & =-{\int }_0^{-\alpha }{\int }_0^{\mathrm{\infty }}{e}^{-t}{e}^{st}dtds\\ & =-{\int }_0^{-\alpha }\mathcal{L}[e^{st}](1)ds\\ & =-{\int }_0^{-\alpha }\frac{ds}{1-s}\\ & =\ln (1+\alpha )\end{array}$$

$$\begin{array}{rl}{\int }_0^{\frac{\pi }{2}}{\tan }^{\alpha }\theta d\theta & ={\int }_0^{\frac{\pi }{2}}{\sin }^{\alpha }\theta {\cos }^{-\alpha }\theta d\theta \\ & ={\int }_0^{\frac{\pi }{2}}{\sin }^{2\frac{\alpha +1}{2}-1}\theta {\cos }^{2\frac{-\alpha +1}{2}-1}\theta d\theta \\ & =\frac{1}{2}\mathrm{B}(\frac{\alpha +1}{2},\frac{-\alpha +1}{2})\\ & =\frac{1}{2}\frac{1}{\mathrm{\Gamma }(1)}\mathrm{\Gamma }\left(\frac{\alpha +1}{2}\right)\mathrm{\Gamma }\left(\frac{-\alpha +1}{2}\right)\\ & =\frac{\pi }{2}\csc (\pi \cdot \frac{\alpha +1}{2})\\ & =\frac{\pi }{2}\sec \frac{\alpha \pi }{2}\end{array}$$

$$\begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(n+\alpha )(n+\beta )}& =\frac{1}{\beta -\alpha }\sum _{n=1}^{\mathrm{\infty }}(\frac{1}{n+\alpha }-\frac{1}{n+\beta })\\ & =\frac{1}{\beta -\alpha }(\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n+\alpha +1}-\sum _{m=0}^{\mathrm{\infty }}\frac{1}{n+\beta +1})\\ & =\frac{\psi (\beta +1)-\psi (\alpha +1)}{\beta -\alpha }\end{array}$$