Tricky improper integral

Calculus Level 4

$\int_0^\infty \dfrac{\sin x}{x+x\sec^2 x}\mathrm dx= \ ?$

\dfrac{\pi(\sqrt{2}-1)}{4\sqrt{2}}

\dfrac{\pi(\sqrt{2}-1)}{\sqrt{2}}

\dfrac{\pi(\sqrt{2}-1)}{2\sqrt{2}}

\dfrac{\pi(\sqrt{2}-1)}{3\sqrt{2}}

1 solution

Nikhil Kumar Singh
Jun 2, 2021

To evaluate this, use Lobachevsky's Integral Formula which says that If $f(x)$ is continuous, $\pi$ -periodic then $\displaystyle\int_0^\infty \dfrac{\sin x}{x}f(x)\mathrm dx=\displaystyle\int_0^{\pi/2} f(x)\mathrm dx$ , so the integral in the question is equal to $\displaystyle\int_0^{\pi/2}\dfrac{1}{1+\sec^2 x}\mathrm dx$ .

Similar and proof of the theorem , here . For more problems of such classes, see the book Table of Integrals, series and products.

Naren Bhandari - 1 week, 1 day ago

Thank you for the book recommendation.

Nikhil Kumar Singh - 1 week, 1 day ago

Tricky improper integral

1 solution

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