Tích phân \(\int\limits^4_0\frac{\text{d}x}{\sqrt{2x+1}+1}\) bằng
\(2-\ln3\). \(2-2\ln2\). \(2-\ln2\). \(4-\ln2\). Hướng dẫn giải:Đặt \(t=\sqrt{2x+1}+1\) \(\Rightarrow x=\frac{\left(t-1\right)^2-1}{2}\)
\(\text{d}x=\left(t-1\right)\text{d}t\)
Đổi cận: \(x|^4_0\Rightarrow t|^4_2\)
\(I=\int\limits^4_0\frac{dx}{\sqrt{2x+1}+1}=\int\limits^4_2\frac{t-1}{t}\text{d}t\)
\(=\int\limits^4_2\left(1-\frac{1}{t}\text{d}t\right)\)
\(=\left(t-\ln t\right)|^4_2=\left(4-\ln4-2+\ln2\right)\)
\(=2-\ln2\).