Lời giải

Đặt \( x = \frac{\pi}{2} - t \Rightarrow dx = -dt \)

\( I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^3 x}{\sin x + \cos x} \, dx = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 t}{\sin t + \cos t} \, dt = J \)

\( I + J = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} \, dx = \int_{0}^{\frac{\pi}{2}} (1 - \sin x \cos x) \, dx\)

            \(  = \left( x - \frac{\sin^2 x}{2} \right) \Bigg|_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - \frac{1}{2}\)

Suy ra: \( I = \frac{\pi}{4} - \frac{1}{4} \)

Bấm:

•  \( \int_{0}^{\frac{\pi}{2}} \frac{\cos^3 x}{\sin x + \cos x} \, dx = 0.535398 \approx \frac{\pi}{4} - \frac{1}{4} \)