30. if $\\int_{-\\infty}^{\\infty} f(x) d x$ is convergent and $a$ and $b$ are real numbers, show that…

Explanation:

Step1: Recall the property of improper integrals

For a convergent improper integral (\int_{-\infty}^{\infty}f(x)dx), we know that (\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{c}f(x)dx+\int_{c}^{\infty}f(x)dx) for any real number (c). Since (\int_{-\infty}^{\infty}f(x)dx) is convergent, we have (\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{a}f(x)dx+\int_{a}^{\infty}f(x)dx) and (\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{b}f(x)dx+\int_{b}^{\infty}f(x)dx).

Step2: Equate the two expressions

Because (\int_{-\infty}^{a}f(x)dx+\int_{a}^{\infty}f(x)dx=\int_{-\infty}^{\infty}f(x)dx) and (\int_{-\infty}^{b}f(x)dx+\int_{b}^{\infty}f(x)dx=\int_{-\infty}^{\infty}f(x)dx), we can set (\int_{-\infty}^{a}f(x)dx+\int_{a}^{\infty}f(x)dx=\int_{-\infty}^{b}f(x)dx+\int_{b}^{\infty}f(x)dx)

Answer:

By the property of convergent improper integrals (\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{c}f(x)dx+\int_{c}^{\infty}f(x)dx) for any real (c), and since (\int_{-\infty}^{\infty}f(x)dx) is convergent, (\int_{-\infty}^{a}f(x)dx+\int_{a}^{\infty}f(x)dx=\int_{-\infty}^{b}f(x)dx+\int_{b}^{\infty}f(x)dx) holds.