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प्रश्न
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
पर्याय
- \[\frac{a + b}{2} \int\limits_a^b f\left( b - x \right) dx\]
- \[\frac{a + b}{2} \int\limits_a^b f\left( b + x \right) dx\]
- \[\frac{b - a}{2} \int\limits_a^b f\left( x \right) dx\]
- \[\frac{b + a}{2} \int\limits_a^b f\left( x \right) dx\]
उत्तर
\[Let\, I = \int_a^b x f\left( x \right) d x .............(1)\]
\[ = \int_a^b \left( a + b - x \right) f\left( a + b - x \right) d x\]
\[ = \int_a^b \left( a + b - x \right) f\left( x \right) dx ...............(2)\]
\[ \text{Adding (1) and (2)}\]
\[2I = \int_a^b \left( x + a + b - x \right) f\left( x \right) d x\]
\[ = \left( a + b \right) \int_a^b f\left( x \right) d x \]
\[Hence\ I = \frac{a + b}{2} \int_a^b f\left( x \right) d x\]
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