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Question about changing an integral bounds due to a reflection.

Suppose you have an integral across $\Bbb R^{+}$ such that $$I=\int_{0}^\infty f(t)dt$$ Would $I$ be the same if I applied a reflection across the y-axis such that $t\to-t$ and integrated along $\Bbb R^{-}$ such that$$I=\int_{-\infty}^0 f(-t)dt$$ Is this ok?

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  • $\begingroup$ @OlivierOloa Thank you for your quick response(+1) $\endgroup$
    – aleden
    Nov 12, 2017 at 17:07

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By the change of variable, $$ u=-t, \qquad dt=-du, \qquad 0 \rightarrow 0,\qquad \infty \rightarrow -\infty, $$ one gets $$ I=\int_{0}^\infty f(t)dt=\int_{0}^{-\infty} f(-u)(-du)=\int_{-\infty}^0 f(-u)du, $$ as announced.

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