Question

example 6 \\(\int \tan^3 x \sec x \\, dx\\)

Explanation:

Step1: Rewrite $\tan^3 x$

$\int \tan^3 x \sec x dx = \int \tan^2 x \cdot \tan x \sec x dx$
Use the identity $\tan^2 x = \sec^2 x - 1$:
$= \int (\sec^2 x - 1) \tan x \sec x dx$

Step2: Substitute $u = \sec x$

Let $u = \sec x$, then $du = \sec x \tan x dx$. Substitute into the integral:
$= \int (u^2 - 1) du$

Step3: Integrate term by term

Integrate each term using the power rule $\int u^n du = \frac{u^{n+1}}{n+1} + C$:
$= \frac{u^3}{3} - u + C$

Step4: Substitute back $u = \sec x$

Replace $u$ with the original variable:
$= \frac{\sec^3 x}{3} - \sec x + C$

Answer:

$\frac{\sec^3 x}{3} - \sec x + C$