Question

Find the value of \(\displaystyle\int_{-1}^{+1}(x+1)\,dx\).

Hint:

Antiderivative is \(x^2/2+x\); or split into an odd part (vanishes) and a constant part.

Answer 1

Step 1: Understanding the Concept:
Integrate \(x+1\) term by term, then apply the limits \(-1\) to \(1\).

Step 2: Finding the antiderivative:
\(\displaystyle\int(x+1)\,dx=\dfrac{x^2}{2}+x\).

Step 3: Applying the limits:
At \(x=1\): \(\dfrac12+1=\dfrac32\). At \(x=-1\): \(\dfrac12-1=-\dfrac12\).

Step 4: Subtracting:
\(\dfrac32-\left(-\dfrac12\right)=2\).

Final Answer:
\(\displaystyle\int_{-1}^{1}(x+1)\,dx=\boxed{2}\).