Question

$\int_{0}^{1}x|x-\frac{1}{2}|dx=$

• A. $1/8$
• B. $1/12$
• C. $1/4$
• D. $1/2$

Hint:

Always split definite integrals at the roots of absolute value functions.

Answer 1

The Correct Option is A

Step 1: Concept
Split the integral at the point where the expression inside the absolute value changes sign ($x = 1/2$).

Step 2: Analysis
- For $x \in [0, 1/2]$, $|x-1/2| = -(x-1/2)$. - For $x \in [1/2, 1]$, $|x-1/2| = (x-1/2)$.

Step 3: Calculation
- $I_1 = \int_{0}^{1/2} (x/2 - x^2) dx = [x^2/4 - x^3/3]_0^{1/2} = 1/16 - 1/24 = 1/48$. - $I_2 = \int_{1/2}^{1} (x^2 - x/2) dx = [x^3/3 - x^2/4]_{1/2}^1 = (1/3 - 1/4) - (1/24 - 1/16) = 1/12 - (-1/48) = 5/48$. - Total $I = 1/48 + 5/48 = 6/48 = 1/8$.

Step 4: Conclusion
Hence, the value is $1/8$. Final Answer: (A)