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