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:
Evaluate ∫₋ₐᵃ |x| dx.

Step 2: Key Formula (Alternate):
|x| is even function. For even f, ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx.

Step 3: Detailed Explanation:
On [0,a], |x|=x. I = 2∫₀ᵃ x dx = 2[x²/2]₀ᵃ = 2(a²/2) = a².

Step 4: Final Answer:
Integral is a².