Q.Evaluate $\displaystyle\int_{0}^{1} e^{x}\,dx$.

The exponential is its own antiderivative, so evaluate $\big[e^x\big]$ between $0$ and $1$.

The antiderivative of $e^x$ is $e^x$ itself (§2).

Evaluate at the limits.

Simplify. Since $e^0 = 1$, the value is $e - 1 \approx 2.718 - 1 = 1.718$.

Check (dual-solve): differentiate the antiderivative back — $\dfrac{d}{dx}(e^x) = e^x$, confirming $e^x$ is the correct antiderivative — and note the value is positive and less than the rectangle of height $e^1 = e$ over width $1$ (area $e \approx 2.718$) yet more …

A frequent error is writing $e^0 = 0$; in fact $e^0 = 1$, so the answer is $e - 1$, not $e$. Another is treating $e^x$ like a power and dividing b …