Question

$\int \frac{sec^8 x}{cosx} dx =$

• A. $\frac{sec^8 x}{8} + C$
• B. $\frac{sec^7 x}{7} + C$
• C. $\frac{sec^6 x}{6} + C$
• D. $\frac{sec^9 x}{9} + C$

Answer 1

Answer: (A)

A

Answer 2

The given integral is $\int \frac{\sec^8 x}{\cos x} dx$.

This simplifies to $\int \sec^8 x \cdot \sec x dx = \int \sec^9 x dx$.

The derivative of options of the form $\frac{\sec^k x}{k} + C$ is $\sec^k x \tan x$.

Since the options do not directly match the integral $\int \sec^9 x dx$, it is highly probable that there is a typo in the question.

Assuming the most common intended form for such problems, if the question meant $\int \sec^8 x \tan x dx$, then by substituting $u = \sec x$, we get $du = \sec x \tan x dx$.

The integral becomes $\int \sec^7 x (\sec x \tan x) dx = \int u^7 du = \frac{u^8}{8} + C = \frac{\sec^8 x}{8} + C$. This matches option (A).