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  • 1.5: Dirichlet’s Tests
    https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.05%3A_Dirichlet%E2%80%99s_Tests
    \[\begin{aligned} \int_{r}^{r_{1}}g(x,y)h(x,y)\,dx&=&\int_{r}^{r_{1}}g(x,y)H_{x}(x,y)\,dx \nonumber\\ &=&g(r_{1},y)H(r_{1},y)-g(r,y)H(r,y)\label{eq:21}\\ &&-\int_{r}^{r_{1}}g_{x}(x,y)H(x,y)\,dx. \nonu...\[\begin{aligned} \int_{r}^{r_{1}}g(x,y)h(x,y)\,dx&=&\int_{r}^{r_{1}}g(x,y)H_{x}(x,y)\,dx \nonumber\\ &=&g(r_{1},y)H(r_{1},y)-g(r,y)H(r,y)\label{eq:21}\\ &&-\int_{r}^{r_{1}}g_{x}(x,y)H(x,y)\,dx. \nonumber\end{aligned}\] \[\begin{aligned} \int_{r}^{r_{1}}\frac{\cos xy}{x+y}\,dx &=& \frac{\sin xy}{y(x+y)}\biggr|_{r}^{r_{1}}+ \int_{r}^{r_{1}}\frac{\sin xy}{y(x+y)^{2}}\,dx\\ &=&\frac{\sin r_{1}y}{y(r_{1}+y)}-\frac{\sin ry}{y(r+y)} +\int_{r}^{r_{1}}\frac{\sin xy}{y(x+y)^{2}}\,dx.\end{aligned}\]

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