\(\int\dfrac{2x+1}{\left(x+1\right)^2+1}dx\)
\(x+1=\tan t\Rightarrow dx=\left(\tan^2t+1\right)dt\)
\(\Rightarrow\int\dfrac{2x+1}{\left(x+1\right)^2+1}dx=\int\dfrac{2\left(\tan t-1\right)+1}{\tan^2t+1}.\left(\tan^2t+1\right)dt\)
\(=\int(2\tan t-1)dt=\int2\tan t.dt-\int dt=2\int\tan t.dt-t\)
\(\int\tan t.dt=\int\dfrac{\sin t}{\cos t}.dt\)
\(u=\cos t\Rightarrow du=-\sin t.dt\Rightarrow\int\dfrac{\sin t}{\cos t}=-\int\dfrac{\sin t}{u}.\dfrac{du}{\sin t}=-ln \left|\cos t\right|+C\)
\(\Rightarrow\int\dfrac{2x+1}{x^2+2x+2}dx=-2ln\left|\cos t\right|-t=-2ln\left|\cos\left[arc\tan\left(x+1\right)\right]\right|-arc\tan\left(x+1\right)\)
P/s: Bạn tự thay cận vô nhé !
\(=\int\limits^1_0\dfrac{2x+2}{x^2+2x+2}dx-\int\limits^1_0\dfrac{1}{\left(x+1\right)^2+1}dx\)
\(=ln\left(x^2+2x+2\right)|^1_0-arctan\left(x+1\right)|^1_0=...\)