Question

\( \int e^x(f(x) + f'(x)) \, dx \) equals:

Hint:

When integrating expressions involving \( e^x \), remember that \( \int e^x f'(x) \, dx = e^x f(x) \).

Answer 1

Step 1: Breaking down the integral.
We are given the integral: \[ I = \int e^x (f(x) + f'(x)) \, dx \] We can split this into two separate integrals: \[ I = \int e^x f(x) \, dx + \int e^x f'(x) \, dx \] This allows us to handle the two terms separately. Step 2: Simplifying the second integral.
We know that the integral of \( e^x f'(x) \) can be simplified using the fundamental theorem of calculus. Specifically, the integral of \( e^x f'(x) \) with respect to \( x \) is simply \( e^x f(x) \). Thus, we have: \[ \int e^x f'(x) \, dx = e^x f(x) \] Step 3: Combining the results.
Now, the first integral is: \[ \int e^x f(x) \, dx = e^x f(x) \] Therefore, the total integral becomes: \[ I = e^x f(x) + C \] Step 4: Conclusion.
Thus, the value of the integral is \( e^x f(x) + C \), where \( C \) is the constant of integration.