Use integration by parts to establish a reduction formula for the integral.
-$$\int \sec ^ { n } x d x , n \neq 1$$
A) $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x - ( n - 2 ) \int \sec ^ { n - 2 } x \tan x d x$
B) $\int \sec ^ { n } x d x = \frac { 1 } { n - 1 } \sec ^ { n - 2 } x \tan x + \frac { n - 2 } { n - 1 } \int \sec ^ { n - 2 } x d x$
C) $\int \sec ^ { n } x d x = \sec ^ { n - 2 } x \tan x + ( n - 2 ) \int \sec ^ { n - 2 } x d x$
D) $\int \sec ^ { n } x d x = \frac { 1 } { n } \sec ^ { n } x \tan x - \frac { n - 1 } { n } \int \sec ^ { n - 1 } x d x$

Question

Quizplus · Accepted Answer

To establish a reduction formula for $\int \sec^n x dx$, we use integration by parts, choosing $u = \sec^{n-2}x$ and $dv = \sec^2x dx$. This leads to the formula $\int \sec^n x dx = \frac{1}{n-1}\sec^{n-2}x\tan x + \frac{n-2}{n-1}\int \sec^{n-2}x dx$, which matches choice B.

Use Integration by Parts to Establish a Reduction Formula for the Integral