Question

The value of \(\int_0^1 x(1-x)^9 \,dx\) is

• A. \(\frac{1}{110}\)
• B. \(\frac{1}{120}\)
• C. \(-\frac{1}{110}\)
• D. \(-\frac{1}{120}\)

Hint:

The property \(\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx\) is extremely useful for definite integrals where the integrand involves a term like \((a-x)^n\). It often simplifies the integrand significantly, making integration straightforward.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to evaluate a definite integral. The integrand has a term \((1-x)^9\) which is difficult to expand.

Step 2: Key Formula or Approach:
We can use a property of definite integrals or substitution.

Property: \(\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx\).

Substitution: Let \(u = 1-x\).
We will use the property as it is often faster.

Step 3: Detailed Explanation:
Let \(I = \int_0^1 x(1-x)^9 \,dx\).
Using the property \(\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx\) with \(a=1\):
Here, \(f(x) = x(1-x)^9\).
So, \(f(a-x) = f(1-x) = (1-x)(1-(1-x))^9 = (1-x)(x)^9 = x^9(1-x)\).
Therefore, the integral becomes:
\[ I = \int_0^1 x^9(1-x) \,dx \]
This is much easier to integrate as we just need to expand the integrand:
\[ I = \int_0^1 (x^9 - x^{10}) \,dx \]
Now, perform the integration:
\[ I = \left[ \frac{x^{10}}{10} - \frac{x^{11}}{11} \right]_0^1 \]
Evaluate at the limits:
\[ I = \left( \frac{1^{10}}{10} - \frac{1^{11}}{11} \right) - \left( \frac{0^{10}}{10} - \frac{0^{11}}{11} \right) \]
\[ I = \left( \frac{1}{10} - \frac{1}{11} \right) - (0) \]
Find a common denominator:
\[ I = \frac{11 - 10}{110} = \frac{1}{110} \]

Step 4: Final Answer:
The value of the definite integral is \(\frac{1}{110}\).