Tích phân \(\int\limits^{\frac{\pi}{2}}_0\frac{\sin x}{\sqrt{1+3\cos x}}\text{d}x\) bằng
\(-\frac{3}{2}\). \(\frac{3}{2}\). \(\frac{2}{3}\). \(-\frac{2}{3}\). Hướng dẫn giải:Đặt \(t=\sqrt{1+3\cos x}\) \(\Rightarrow1+3\cos x=t^2\)
\(-3\sin x\text{d}x=2t\text{d}t\)
Đổi cận:
\(x|^{\frac{\pi}{2}}_0\Rightarrow t|^1_2\)
\(I=\int\limits^{\frac{\pi}{2}}_0\frac{\sin x}{\sqrt{1+3\cos x}}\text{d}x=\int\limits^1_2\frac{-\frac{2}{3}t\text{d}t}{t}=\frac{2}{3}\int\limits^2_1\text{d}t\)
\(=\frac{2}{3}t|^2_1=\frac{4}{3}-\frac{2}{3}=\frac{2}{3}\).