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-$\text { A student knows that } \int _ { a } ^ { + \infty } f ( x ) d x \text { converges. Does } \int _ { - \infty } ^ { a } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text { also necessarily converge? }$

Question

Quizplus · Accepted Answer

Knowing that $\int_a^{+\infty} f(x)dx$ converges does not guarantee that $\int_{-\infty}^a f(x)dx$ converges. For example, consider the function $f(x)=\frac{1}{x^2+1}$. We have $\int_0^{+\infty} \frac{dx}{x^2+1}<+\infty$, but $\int_{-\infty}^0 \frac{dx}{x^2+1}=+\infty$.

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