Q86 / 180 · COMEDK 2023 (Evening Shift) · Mathematics
$\int e^x\left(1+\tan x+\tan ^2 x\right) d x \text { is equal to }$
- A$e^x \sin x+c$
- B$e^x \sec x+c$
- C$e^x \cos x+c$
- Correct answerD$e^x \tan x+c$
Solution
The integral is $I = \int e^x (1 + \tan x + \tan^2 x) dx$.
Using the identity $1 + \tan^2 x = \sec^2 x$, the expression becomes $I = \int e^x (\sec^2 x + \tan x) dx$.
Let $f(x) = \tan x$. Then $f'(x) = \sec^2 x$.
The integral is in the form $\int e^x (f(x) + f'(x)) dx$, which evaluates to $e^x f(x) + c$.
Therefore, $I = e^x \tan x + c$.
Answer: $e^x \tan x+c$
Using the identity $1 + \tan^2 x = \sec^2 x$, the expression becomes $I = \int e^x (\sec^2 x + \tan x) dx$.
Let $f(x) = \tan x$. Then $f'(x) = \sec^2 x$.
The integral is in the form $\int e^x (f(x) + f'(x)) dx$, which evaluates to $e^x f(x) + c$.
Therefore, $I = e^x \tan x + c$.
Answer: $e^x \tan x+c$
Hence, the correct answer is option: 'D'.