If ^1_0f(x)dx=1, ^1_0xf(x)dx=a, ^1_0x^2f(x)dx=a^2, then ^1_0(a-x)… - Vidyadip

Question

If $\int\limits^1_0\text{f}(\text{x})\text{dx}=1,\int\limits^1_0\text{x}\text{f}(\text{x})\text{dx}=\text{a},\int\limits^1_0\text{x}^2\text{f}(\text{x})\text{dx}=\text{a}^2,$ then $\int\limits^1_0(\text{a}-\text{x})^2\text{f(x)}\text{dx}$ equals:

4a²
0
2a²
none of these

✓

Answer

$0$

Solution:

$\int\limits^1_0(\text{a}-\text{x})^2\text{ f}(\text{x})\text{dx}$

$=\text{a}^2\int\limits^1_0\text{f}(\text{x})\text{dx}+\int\limits^1_0\text{x}^2\text{f}(\text{x})\text{dx}-2\text{a}\int\limits^1_0\text{x}\text{f}(\text{x})\text{dx}$

$=\text{a}^2\times1+\text{a}^2-2\text{aa}$ (As per given values)

$=2\text{a}^2-2\text{a}^2$

$=0$

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