Question

AYUDA PORFAVOR ES URGENTE AL QUE NO SABE NO RESPONDER LE REPORTO​

Answer 1

A ver intentemos

sea x = sec t , entonces x² - 1 = tan²t

luego x³ √(x²-1) = sec³t · tan t

Por otra parte dx = sec t tan t dt

Entonces

$\displaystyle\\I = \int_{1}^{+\infty}\dfrac{1}{x^3\sqrt{x^2-1}}~dx\\ ~\\I=\int_{0}^{\pi/2}\dfrac{\sec t \tan t}{\sec^3 \tan t}~dt\\~\\I=\int_{0}^{\pi/2} \cos^2 t ~dt\\~\\\\I=\dfrac{1}{2}\int_{0}^{\pi/2} 1+\cos (2t) ~dt\\~\\I=\left.\dfrac{t}{2} + \dfrac{\sin(2t)}{4} \right|_{0}^{\pi/2}\\~\\\\\boxed{I=\dfrac{\pi}{4}}$