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If F is an Integrable Function Such that F(2a − X) = F(X), Then Prove that 2 a ∫ 0 F ( X ) D X = 2 a ∫ 0 F ( X ) D X - Mathematics

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प्रश्न

If `f` is an integrable function such that f(2a − x) = f(x), then prove that

\[\int\limits_0^{2a} f\left( x \right) dx = 2 \int\limits_0^a f\left( x \right) dx\]

 

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उत्तर

\[Let\ I = \int_0^{2a} f\left( x \right) d x\]
\[\text{By Additive property}\]
\[I = \int_0^a f\left( x \right) d x + \int_a^{2a} f\left( x \right) d x\]
\[\text{Consider the integra}l \int_a^{2a} f\left( x \right) d x\]
\[Let\ x = 2a - t, \text{then }dx = - dt\]
\[When\ x = a, t = a, x = 2x, t = 0\]
\[\text{Hence } \int_a^{2a} f\left( x \right) d x = - \int_a^0 f\left( 2a - t \right) d t\]
\[ = \int_0^a f\left( 2a - t \right) d t\]
\[ = \int_0^a f\left( 2a - x \right) dx ...............\left( \text{Changing the variable} \right)\]
Therefore,
\[I = \int_0^a f\left( x \right) d x + \int_0^a f\left( 2a - x \right) d x\]
\[ = \int_0^a f\left( x \right) d x + \int_0^a f\left( x \right) d x .................\left[\text{Given }\int_0^a f\left( x \right) d x = \int_0^a f\left( 2a - x \right) d x \right]\]
\[ = 2 \int_0^a f\left( x \right) d x\]

Hence Proved.

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Definite Integrals
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अध्याय 20: Definite Integrals - Exercise 20.5 [पृष्ठ ९६]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 20 Definite Integrals
Exercise 20.5 | Q 43 | पृष्ठ ९६

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