Question

For a real number \(a\), let \(I(a) = \displaystyle\int_{-1}^{1} (3x^2 - ax + 1)\, dx\). Which of the following statements is/are true?

• A. The value of \(I(a)\) is independent of the value of \(a\)
• B. The value of \(I(a)\) can vary with the value of \(a\)
• C. There exists \(a \in (-\infty, +\infty)\) such that \(I(a)\) is a positive real number
• D. There exists \(a \in (-\infty, +\infty)\) such that \(I(a)\) is a negative real number

Hint:

The term with a is an odd power of x over a symmetric interval, so it always integrates to zero; only the even-power terms survive.

Answer 1

The Correct Option is A, C

Step 1: Split the integral into three parts.
\[ I(a) = \int_{-1}^{1} 3x^2\, dx \;-\; \int_{-1}^{1} ax\, dx \;+\; \int_{-1}^{1} 1\, dx \]
We evaluate each part separately.

Step 2: Evaluate the first part.
\[ \int_{-1}^{1} 3x^2\, dx = \Big[x^3\Big]_{-1}^{1} = 1^3 - (-1)^3 = 1 - (-1) = 2 \]

Step 3: Evaluate the second part using symmetry.
The function \(ax\) is an odd function of \(x\), since replacing \(x\) with \(-x\) flips its sign, and the interval \([-1,1]\) is symmetric about \(0\). The integral of any odd function over a symmetric interval is always \(0\). So
\[ \int_{-1}^{1} ax\, dx = 0 \quad \text{for every value of } a \]

Step 4: Evaluate the third part.
\[ \int_{-1}^{1} 1\, dx = \Big[x\Big]_{-1}^{1} = 1 - (-1) = 2 \]

Step 5: Combine the parts.
\[ I(a) = 2 - 0 + 2 = 4 \]
This value does not contain \(a\) anywhere, so \(I(a) = 4\) for every choice of \(a\).

Step 6: Check each statement.

(A) Independent of \(a\): Since \(I(a) = 4\) regardless of \(a\), this is true.

(B) Varies with \(a\): Since \(I(a)\) is always \(4\), it never changes, so this is false.

(C) There exists \(a\) making \(I(a)\) positive: \(I(a) = 4\) is positive for every \(a\), so such an \(a\) certainly exists. This is true.

(D) There exists \(a\) making \(I(a)\) negative: \(I(a) = 4\) is never negative for any \(a\), so this is false.

Final Answer:
\(I(a) = 4\) for all real \(a\), so statements (A) and (C) are true.
\[ \boxed{I(a) = 4 \text{ for all } a} \]