Question

If \(\int e^x(tanx+1)secxdx=e^xf(x)+C,\) then \(f(x)\) is:
(A) \(e^X\)
(B) \(tanx\)
(C)\(secx\)
(D) \(secx \ tanx\)
choose the correct answer from the options given below

 

• (A) Only
• (B) Only
• (C) Only
• (D) Only

Answer 1

The Correct Option is C

The given integral is: \(\int e^x(\tan x + 1) \sec x \, dx = e^x f(x) + C\). We need to find \(f(x)\).

Start by simplifying the integral: \(\int e^x (\tan x + 1) \sec x \, dx\).

We can split the integral into two parts: \(\int e^x \tan x \sec x \, dx + \int e^x \sec x \, dx\).

Consider the first integral, \(\int e^x \tan x \sec x \, dx\):

- Notice that \(\tan x \sec x\) is the derivative of \(\sec x\).

Thus, this becomes \(\int e^x d(\sec x)\), which results in \(e^x \sec x\).

Next, consider the second integral, \(\int e^x \sec x \, dx\):

- This integral by itself does not result in a simple elementary function solution but it is already covered in the derivative approach.

We find that both components actually map to the same term \(e^x \sec x\) under an integrated form.

Thus, combining both components, \(f(x)\) must therefore be \(secx\).

Therefore, the correct answer is: (C) Only