Which expression is an antiderivative of e^x with respect to x?

Which expression is an antiderivative of e^x with respect to x?

• A. e^{x} + C
• B. e^{2x} + C
• C. x e^{x} + C
• D. e^{−x} + C

Answer: (A)

The key idea is that e^x is its own derivative, so its antiderivative must be e^x plus a constant. Since d/dx [e^x] = e^x, we have ∫ e^x dx = e^x + C. For e^{2x}, differentiating (1/2) e^{2x} gives e^{2x}, so its antiderivative is (1/2) e^{2x} + C, not e^{2x}. For x e^{x}, using the product rule gives d/dx [x e^x] = e^x + x e^x, which is not just e^x. For e^{−x}, the derivative is −e^{−x}, so its antiderivative would be −e^{−x} + C. Therefore, the correct antiderivative is e^x + C.

The key idea is that e^x is its own derivative, so its antiderivative must be e^x plus a constant. Since d/dx [e^x] = e^x, we have ∫ e^x dx = e^x + C.

For e^{2x}, differentiating (1/2) e^{2x} gives e^{2x}, so its antiderivative is (1/2) e^{2x} + C, not e^{2x}. For x e^{x}, using the product rule gives d/dx [x e^x] = e^x + x e^x, which is not just e^x. For e^{−x}, the derivative is −e^{−x}, so its antiderivative would be −e^{−x} + C.

Therefore, the correct antiderivative is e^x + C.