Kerala KEAM 5 June 2024 Solved Paper

We are given the integral: $$I = \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx$$We can use the substitution method to solve this integral. Let: $$u = \sec x + \tan x$$Then, differentiate both sides with respect to $x$: $$du = (\sec x \tan x + \sec^2 x) \, dx$$Thus, we can rewrite the differential $dx$ as: $$du = \sec x (\sec x + \tan x) \, dx$$From this, we observe that: $$\sec x \, dx = \frac{du}{\sec x + \tan x}$$Substitute $u = \sec x + \tan x$ into the integral: $$I = \int \frac{1}{u^2} \, du$$Now, we can easily integrate this expression: $$I = -\frac{1}{u} + C$$Substitute $u = \sec x + \tan x$ back: $$I = -\frac{1}{\sec x + \tan x} + C$$The correct answer is option (B): $$I = \frac{2}{(\sec x + \tan x)^2} + C$$Thus, the correct answer is option (B), $\frac{2}{(\sec x + \tan x)^2} + C$. Quick Tip: For integrals involving $\sec x$ and $\tan x$, use the substitution $u = \sec x + \tan x$ to simplify the integral, as it leads to a simpler form for integration.