e^x ( x + _e x) dx = Option 1: e^x _e x + C Option 2: _e x + C Option 3: e^x x + C Option 4: e^x x + C

Answer: e^x _e x + C. Solution: Let f(x)=_e x. Then f'(x)=( x x)/( x)= x. The integrand is e^x[f'(x)+f(x)]. Using e^x[f(x)+f'(x)]dx=e^x f(x)+C, the result is e^x_e x+C.

Calculus

Integrals

Medium

core

$\int e^x (\tan x + \log_e \sec x) \, dx =$

$e^x \log_e \sec x + C$

$\log_e \sec x + C$

$e^x \tan x + C$

$e^x \sec x + C$

✅ Correct Option: 1

Let $f(x)=\log_e\sec x$ . Then $f'(x)=\frac{\sec x\tan x}{\sec x}=\tan x$ .

The integrand is $e^x[f'(x)+f(x)]$ .

Using $\int e^x[f(x)+f'(x)]dx=e^x f(x)+C$ , the result is $e^x\log_e\sec x+C$ .

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$\int e^x (\tan x + \log_e \sec x) \, dx =$