Cho hàm số f(x) liên tục trên \(\mathbb{R}\) và\(f...

C

- Hướng dẫn giải

Đặt \(x=-t\)

Với \(x=\frac{-\pi}{2}\Rightarrow t=\frac{\pi}{2}\)

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

Khi đó

\(\int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} f(x) d x=\int\limits_{\frac{\pi}{2}}^{-\frac{\pi}{2}} f(-t)(-d t)=\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(-t) d t=\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(-x) d x\)

\(\Rightarrow 2 \int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} f(x) d x=\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}[f(x)+f(-x)] d x=\int\limits_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos ^{4} x d x \Rightarrow I=\frac{3 \pi}{16}\)