Lời giải

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

\( I = \int_0^{\frac{\pi}{2}} \frac{5 \cos x - 4 \sin x}{(\cos x + \sin x)^3} \, dx = \int_0^{\frac{\pi}{2}} \frac{5 \sin t - 4 \cos t}{(\sin t + \cos t)^3} \, dt\)

   \( = \int_0^{\frac{\pi}{2}} \frac{5 \sin x - 4 \cos x}{(\sin x + \cos x)^3} \, dx = J\)
 

\(I + J = \int_0^{\frac{\pi}{2}} \frac{\cos x + \sin x}{(\cos x + \sin x)^3} \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{(\cos x + \sin x)^2} \, dx\)

           \( = \int_0^{\frac{\pi}{2}} \frac{1}{2 \cos^2 ( x - \frac{\pi}{4}) } \, dx = \frac{1}{2} tan \left( x - \frac{\pi}{4} \right) \Big|_0^{\frac{\pi}{2}}\)

           \(= 1\)

Suy ra:  \(I = \frac{1}{2}\)