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: Understanding the Question:
To integrate a function with an absolute value $|f(x)|$, split the integral at the points where $f(x)$ changes sign.

Step 3: Detailed Explanation:
$|x - 1/2|$ is $-(x - 1/2)$ for $x < 1/2$ and $(x - 1/2)$ for $x > 1/2$.
$I = \int_0^{1/2} x(1/2 - x) dx + \int_{1/2}^1 x(x - 1/2) dx$ $I = \int_0^{1/2} (x/2 - x^2) dx + \int_{1/2}^1 (x^2 - x/2) dx$
$I = \left[ \frac{x^2}{4} - \frac{x^3}{3} \right]_0^{1/2} + \left[ \frac{x^3}{3} - \frac{x^2}{4} \right]_{1/2}^1$
$I = (\frac{1}{16} - \frac{1}{24}) + [ (\frac{1}{3} - \frac{1}{4}) - (\frac{1}{24} - \frac{1}{16}) ]$
$I = \frac{3-2}{48} + \frac{4-3}{12} - \frac{2-3}{48}$ $I = \frac{1}{48} + \frac{4}{48} + \frac{1}{48} = \frac{6}{48} = \frac{1}{8}$.

Step 4: Final Answer:
The value of the integral is $1/8$.