【高校数学】今週の積分

\[\begin{eqnarray*} (No.1) \\ \int \tan ^3 x dx &=& \int \tan x\cdot\tan ^2 x dx\\ &=& \int \tan x\cdot\left(\frac{1}{\cos^2x}-1\right) dx\\ &=& \int \tan x\cdot\frac{1}{\cos^2x}dx+\int \frac{(-\sin x)}{\cos x} dx\\ &=& \frac{1}{2}\tan^2x + \log\left|\cos x\right|+C\\ \\ I_n &=& \int \tan ^n x dx\\ &=& \int \tan ^{n-2} x\cdot \tan ^2 x dx\\ &=& \frac{1}{n-1} \tan ^{n-1} - I_{n-2}\\ \\ (No.2) \\ \int_{0}^{2\pi}\sqrt{1+\cos x} dx\\ \end{eqnarray*}\]