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

Question

Quizplus · Accepted Answer

To derive the reduction formula for $\int \tan^n x dx$, we use integration by parts, letting $u = \tan^{n-1} x$ and $dv = \tan x dx$. This leads to $du = (n-1)\tan^{n-2}x\sec^2x dx$ and $v = -\ln|\cos x|$. Applying integration by parts formula, we simplify to get $\int \tan^n x dx = \frac{1}{n-1}\tan^{n-1}x - \int \tan^{n-2}x dx$, which matches choice A.

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