Question

The value of $$ \int_0^1 x^4 (1 - x)^2 \, dx $$ is:

• A. \( \frac{1}{105} \)
• B. \( \frac{1}{501} \)
• C. \( \frac{1}{205} \)
• D. \( \frac{1}{502} \)

Hint:

Break complex integrals by expansion and integrate term-by-term.

Answer 1

The Correct Option is A

Expand the integrand: \[ (1 - x)^2 = 1 - 2x + x^2 \] So, \[ x^4 (1 - x)^2 = x^4 - 2x^5 + x^6 \] Integrate term-wise: \[ \int_0^1 x^4 \, dx = \frac{1}{5}, \quad \int_0^1 x^5 \, dx = \frac{1}{6}, \quad \int_0^1 x^6 \, dx = \frac{1}{7} \] Thus, \[ \int_0^1 x^4 (1-x)^2 \, dx = \frac{1}{5} - 2 \times \frac{1}{6} + \frac{1}{7} = \frac{1}{5} - \frac{1}{3} + \frac{1}{7} \] Find common denominator 105: \[ \frac{21}{105} - \frac{35}{105} + \frac{15}{105} = \frac{21 - 35 + 15}{105} = \frac{1}{105} \]