Question

Calculate \[ \int e^x\,dx. \]

• A. \( e^x+C \)
• B. \( e^{x+1}+C \)
• C. \( \frac{e^x}{x}+C \)
• D. \( xe^x+C \)

Hint:

Remember: \[ \int e^x\,dx=e^x+C \] because \(e^x\) is its own derivative.

Answer 1

The Correct Option is A

Concept: Integration is the reverse process of differentiation. The exponential function \(e^x\) is unique because its derivative is equal to itself: \[ \frac{d}{dx}(e^x)=e^x \] Therefore, its integral is also itself.

Step 1: Recall the standard integration formula \[ \int e^x\,dx=e^x+C \] where \(C\) is the constant of integration.

Step 2: Verify by differentiation Differentiate the obtained answer: \[ \frac{d}{dx}(e^x+C) \] \[ =e^x+0 \] \[ =e^x \] which matches the original integrand. Hence the result is correct. Final Answer: \[ \boxed{e^x+C} \]