Q.Evaluate $\displaystyle\int_{1}^{2} (x^{2} + 2x)\,dx$.

Integrate term by term and evaluate from $1$ to $2$.

$\big[\tfrac{x^3}{3}+x^2\big]_1^2 = (\tfrac83+4)-(\tfrac13+1) = \tfrac{20}{3}-\tfrac43 = \tfrac{16}{3}$.

Use linearity and the power rule.

Antiderivative. $\displaystyle\int (x^2 + 2x)\,dx = \frac{x^3}{3} + 2\cdot\frac{x^2}{2} = \frac{x^3}{3} + x^2$.

Evaluate at the limits.

Check (dual-solve): integrate the two terms separately. $\displaystyle\int_1^2 x^2\,dx = \left[\tfrac{x^3}{3}\right]_1^2 = \tfrac{8-1}{3} = \tfrac{7}{3}$; $\displaystyle\int_1^2 2x\,dx = \big[x^2\big]_1^2 = 4 - 1 = 3 = \tfrac{9}{3}$. Adding: $\tfrac{7}{3} + \tfrac{9}{3} = \tfrac{16}{3}$, matching.

A common error is careless fraction arithmetic — convert whole numbers to thirds ($4 = \tfrac{12}{3}$, $1 = \tfrac{3}{3}$) before combining, or the terms will not add correctly. Also apply the power rule to each term: $\int x^2\,dx = \tfrac{x^3}{3}$, not $\tfrac{x^3}{2}$.