Question

Evaluate $\displaystyle \int_{0}^{1} x(1-x)^5 \, dx$

• A. $\frac{1}{7}$
• B. $-\frac{1}{42}$
• C. $\frac{1}{42}$
• D. $\frac{1}{6}$

Hint:

For integrals involving $(1-x)^n$, substitution $u = 1-x$ often simplifies the calculation.

Answer 1

The Correct Option is C

Step 1: Using substitution.
Let $u = 1 - x$, then $du = -dx$. When $x = 0,\ u = 1$ and when $x = 1,\ u = 0$.
Step 2: Changing the limits and integral.
\[ \int_{0}^{1} x(1-x)^5 dx = \int_{1}^{0} (1-u)u^5(-du) \] \[ = \int_{0}^{1} (1-u)u^5 du \]
Step 3: Expanding and integrating.
\[ \int_{0}^{1} (u^5 - u^6) du \] \[ = \left[\frac{u^6}{6} - \frac{u^7}{7}\right]_0^1 \] \[ = \frac{1}{6} - \frac{1}{7} = \frac{1}{42} \]
Step 4: Conclusion.
\[ \int_{0}^{1} x(1-x)^5 dx = \frac{1}{42} \]