Answer to Question #175012 in Calculus for Phyroe

Question

∫ sec^n β tan β dβ

Answer 1

$=\int \frac{\sin \left(β\right)}{\cos ^{n+1}\left(β\right)}dβ$

$=\int \frac{\sin \left(β\right)}{\cos ^{n+1}\left(β\right)}dβ$

$=\int \:-\frac{1}{u^{n+1}}du$

$\frac{1}{u^{n+1}}=u^{-n-1}$

$=-\int \:u^{-n-1}du$

$=-\frac{u^{-n-1+1}}{-n-1+1}$

$=-\frac{\cos ^{-n-1+1}\left(β\right)}{-n-1+1}$

$=\frac{1}{n}\cos ^{-n}\left(β\right)$

$=\frac{1}{n}\cos ^{-n}\left(β\right)+C$