AP EAMCET · Maths · Definite Integration
If \(\int_0^1 f(x) d x=1, \int_0^1 x f(x) d x=a\) and \(\int_0^1 x^2 f(x) d x=a^2\), then \(\int_0^1(x-a)^2 f(x) d x\) is equal to
- A \(a^2\)
- B \(a^2+1\)
- C \(a^2-1\)
- D 0
Answer & Solution
Correct Answer
(D) 0
Step-by-step Solution
Detailed explanation
\[ \text { } \begin{aligned} & \int_0^1(x-a)^2 f(x)=\int_0^1\left(x^2 f(x)+a^2 f(x)-2 a x f(x)\right) d x \\ & =\int_0^1 x^2 f(x) d x+a^2 \int_0^1 f(x) d x-2 a \int_0^1 x f(x) d x \\ & =a^2+a^2(1)-2 a(a) \\ & =a^2+a^2-2 a^2=0 \end{aligned} \]
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