Evaluate $\displaystyle\int_{2}^{4} (x + 3)\,dx$.

Question

Accepted Answer

Apply the Fundamental Theorem.

**Antiderivative.** $\displaystyle\int (x+3)\,dx = \frac{x^2}{2} + 3x$. Take $F(x) = \dfrac{x^2}{2} + 3x$.

**Evaluate at the limits.**
$$\int_{2}^{4}(x+3)\,dx = \left[\frac{x^2}{2} + 3xight]_{2}^{4} = \left(\frac{16}{2} + 12ight) - \left(\frac{4}{2} + 6ight) = (8 + 12) - (2 + 6) = 20 - 8 = 12.$$

**Check (dual-solve):** $y = x + 3$ is linear and positive on $[2,4]$, so the region is a trapezium with parallel ordinates $f(2) = 5$ and $f(4) = 7$ and width $4 - 2 = 2$: area $= 	frac12(5+7)(2) = 	frac12(12)(2) = 12$, matching.
> [!ANSWER]
> $\displaystyle\int_{2}^{4}(x+3)\,dx = 12$.

Q.Evaluate $\displaystyle\int_{2}^{4} (x + 3)\,dx$ .

Concept understanding — Definite Integral and the Fundamental Theorem

Q.Evaluate ∫24(x+3) dx\displaystyle\int_{2}^{4} (x + 3)\,dx∫24​(x+3)dx.

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Q.Evaluate $\displaystyle\int_{2}^{4} (x + 3)\,dx$ .

Unlock everything free for 14 days