allery$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:
A$f(x) f'(x) + C$
B$e^x f(x) + C$
C$e^x f'(x) + C$
D$e^x\bigl(f(x) - f'(x)\bigr) + C$
Answer & Solution
Correct answer: B. $e^x f(x) + C$
This is the standard 'product with exponential' identity: $\dfrac{d}{dx}\bigl(e^x f(x)\bigr) = e^x f(x) + e^x f'(x) = e^x(f(x) + f'(x))$
so by anti-differentiation the integral is $e^x f(x) + C$.
A useful trick whenever an integrand splits as $e^x$ times (something + its derivative).
Keep practising Unit 6A: Indefinite Integrals & Fundamental Theorem
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