Question

If an antiderivative of \(f(x)\) is \(e^x\) and an antiderivative of \(g(x)\) is \(cosx\), then \(\int f(x)cosx\,dx+\int g(x)e^x\,dx =\)

• A. \(e^xsinx+c\)
• B. \(e^x(f(x)+g(x))+c\)
• C. \(e^xcosx+c\)
• D. \(e^x+c\)

Hint:

The two integrands are the parts of d(e^x cos x).

Answer 1

The Correct Option is C

Step 1: Product rule:
$\dfrac{d}{dx}(e^x\cos x)=e^x\cos x+e^x(-\sin x)$.

Step 2: Match terms:
The first term is $f\cos x$ since $f=e^x$. The second term is $g\,e^x$ since $g=-\sin x$.

Step 3: Conclude:
So the sum of the integrals is the antiderivative of that derivative, which is $e^x\cos x+c$. Option (C).

Final Answer:
The two integrands add up to the derivative of e^x cos x. \[ \boxed{C} \]