The value of \(\int_{0}^{1}\displaystyle e^x(x+1)\,dx\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(e\)
The integrand has the form \(e^x\left(f(x) + f'(x)\right)\) with \(f(x) = x\), \(f'(x) = 1\). By the standard result
\( \displaystyle \int e^x\left(f(x) + f'(x)\right)dx = e^x f(x) + C \)
the antiderivative is \(xe^x\). (Check: \(\frac{d}{dx}(xe^x) = e^x + xe^x = e^x(x+1)\).)
\( \displaystyle \int_{0}^{1} e^x(x+1)\,dx = \Big[xe^x\Big]_{0}^{1} \)
\(= 1 \cdot e - 0 = e\)