7.1 Integration By Parts/50

Prove $\int _{}^{}\sec ^{n}(x)dx={\frac {\tan(x)\cdot \sec ^{n-2}(x)}{n-1}}+{\frac {n-2}{n-1}}\int _{}^{}\sec ^{n-2}(x)dx$

$\int _{}^{}\sec ^{n}(x)dx=\int _{}^{}\sec ^{2}(x)\cdot \sec ^{n-2}(x)dx$

${\begin{aligned}&u=\sec ^{n-2}(x)\quad &dv=\sec ^{2}(x)dx\\[2ex]&du=(n-2)\sec ^{n-3}\cdot \sec(x)\tan(x)dx\quad &v=\tan(x)\\[2ex]\end{aligned}}$

${\begin{aligned}\int _{}^{}\sec ^{2}(x)\cdot \sec ^{n-2}(x)dx&=\sec ^{n-2}(x)\cdot \tan(x)-\int _{}^{}\left[(n-2)\sec ^{n-3}(x)\cdot \sec(x)\tan(x)\right]\cdot \tan(x)dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\left[\sec ^{n-2}(x)\cdot \tan ^{2}(x)\right]dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\sec ^{n-2}(x)\cdot [\sec ^{2}(x)-1]dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\left[\sec ^{n}(x)-\sec ^{n-2}(x)\right]dx\\[2ex]\end{aligned}}$

${\begin{aligned}\int _{}^{}\sec ^{n}(x)dx=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\sec ^{n}(x)dx+(n-2)\int _{}^{}\sec ^{n-2}(x)dx\end{aligned}}$

${\begin{aligned}&+(n-2)\int _{}^{}\sec ^{2}(x)dx\quad &&&+(n-2)\int _{}^{}\sec ^{2}(x)dx\end{aligned}}$

${\begin{aligned}(n-1)\int _{}^{}\sec ^{n}(x)dx=\sec ^{2}(x)\tan(x)+(n-2)\int _{}^{}\sec ^{n-2}(x)dx\\[2ex]&={\frac {\sec ^{n-2}(x)\tan(x)}{n-1}}+{\frac {n-2}{n-1}}\int _{}^{}\sec ^{n-2}(x)dx\\[2ex]\end{aligned}}$

Note: ${\begin{aligned}\tan ^{2}(x)=\sec ^{2}(x)-1\end{aligned}}$