if-displaystyle-int_-inftyinfty-fx-dx-is-convergent-and-a-and-b-are-real-numbers-show-that-int_-inft

Question

If $ \displaystyle \int_{-\infty}^\infty f(x)\ dx $ is convergent and $ a $ and $ b $ are real numbers, show that
$$ \int_{-\infty}^a f(x)\ dx + \int_a^\infty f(x)\ dx = \int_{-\infty}^b f(x)\ dx + \int_b^\infty f(x)\ dx $$

Wen Zheng · Accepted Answer

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Problem

Show that $ \displaystyle \int_0^\infty x^2 e^{-x…

Problem 76 Hard Difficulty

If $ \displaystyle \int_{-\infty}^\infty f(x)\ dx $ is convergent and $ a $ and $ b $ are real numbers, show that
$$ \int_{-\infty}^a f(x)\ dx + \int_a^\infty f(x)\ dx = \int_{-\infty}^b f(x)\ dx + \int_b^\infty f(x)\ dx $$

Answer

$\int_{0}^{\infty} f(x) d x$

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