[tex]\int\limits_{-\infty}^{+\infty}e {}^{ - x} \: dx \: = \: \int\limits_{-\infty}^{0}e {}^{ - x} \: dx + \int\limits_{0}^{+\infty}e {}^{ - x} \: dx \\ \\ u = - x \: \to \: \: \frac{du}{dx} = - 1 \: \: \to \: \: dx = - du \\ \\ \: \ \lim_{b \to - \infty } \int\limits_{b}^{0}e {}^{ u} \: ( - du) + \lim_{c \to \infty } \int\limits_{0}^{c}e {}^{ u} \: ( - du) \\ \\ \lim_{b \to - \infty }( - e {}^{ - x}) _{b}^{0}+ \lim_{c \to \infty } ( - e {}^{ - x} )_{0}^{c} \\ \\ \lim_{b \to - \infty }( - e {}^{ - 0} +e {}^{ - b} ) + \lim_{c \to \infty } ( - e {}^{ - c} + e {}^{ - 0} )_{0}^{c} \\ \\ \lim_{b \to - \infty }( - 1 + \infty ) + \lim_{c \to \infty } ( 0 + 1) \\ \\ \infty + 1= \infty \: (diverge)[/tex]
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[tex]\int\limits_{-\infty}^{+\infty}e {}^{ - x} \: dx \: = \: \int\limits_{-\infty}^{0}e {}^{ - x} \: dx + \int\limits_{0}^{+\infty}e {}^{ - x} \: dx \\ \\ u = - x \: \to \: \: \frac{du}{dx} = - 1 \: \: \to \: \: dx = - du \\ \\ \: \ \lim_{b \to - \infty } \int\limits_{b}^{0}e {}^{ u} \: ( - du) + \lim_{c \to \infty } \int\limits_{0}^{c}e {}^{ u} \: ( - du) \\ \\ \lim_{b \to - \infty }( - e {}^{ - x}) _{b}^{0}+ \lim_{c \to \infty } ( - e {}^{ - x} )_{0}^{c} \\ \\ \lim_{b \to - \infty }( - e {}^{ - 0} +e {}^{ - b} ) + \lim_{c \to \infty } ( - e {}^{ - c} + e {}^{ - 0} )_{0}^{c} \\ \\ \lim_{b \to - \infty }( - 1 + \infty ) + \lim_{c \to \infty } ( 0 + 1) \\ \\ \infty + 1= \infty \: (diverge)[/tex]