\( L ^ { - 1 } ( \operatorname { Ln } \frac { s ^ { 2 } + 1 } { s ( s + 1 ) } ) \)

\( y=\sinh (x) \)

\( \int _ { 0 } ^ { \infty } e ^ { x ^ { 2 } } d x \)

\( \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } e ^ { \frac { x } { y } } d y d x \)

put \( z=\mathrm{Re}^{i \theta} \), then \( |z|=R \) Now, \( \left|\int_{\Gamma} f(z) d z\right| \leq \frac{\pi R}{\left(R^{2}-1\right)^{2}} \rightarrow 0 \) as \( R \rightarrow \infty \) Thus, the equation (1) becomes \( \lim _{R \rightarrow \infty} \int_{-R}^{R} f(x) d x+\lim _{R \rightarrow \infty} \int_{\Gamma} f(z) d z=\frac{\pi}{2} \) \( \Rightarrow \quad \int_{-\infty}^{\infty} f(x) d x+0=\frac{\pi}{2} \Rightarrow \int_{-\infty}^{\infty} f(x) d x=\frac{\pi}{2} \)

\( \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { e ^ { - y } } { y } d y d x \)