Question

Evaluate \[ I=\int_{-2}^{0} \Big(x^3+3x^2+3x+3+(x+1)\cos(x+1)\Big)\,dx \]

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\) \bigskip

Hint:

For definite integrals over symmetric limits \([-a,a]\): \[ \int_{-a}^{a}\text{(odd function)}\,dx=0. \] After the substitution \(t=x+1\), the odd terms vanish immediately, leaving only the constant term.

Answer 1

The Correct Option is A

Step 1: Substitute \(t = x+1\). \(x^3+3x^2+3x+3 = (x+1)^3+2 = t^3+2\). Also \(\cos(x+1) = \cos t\). Limits: \(x=-2 \to t=-1\), \(x=0 \to t=1\). Integral: \(I = \int_{-1}^1 (t^3 + t\cos t + 2)dt\).

Step 2: \(t^3\) is odd, \(t\cos t\) is odd, both integrate to 0 over symmetric interval. \(I = \int_{-1}^1 2 dt = 4\).

Step 3: Write the final answer. \(\boxed{4}\)