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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
If f(a + b - x) = f (x), then $\int_{a}^{b} x f(x) d x$ is equal to
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Answer

Given Integral is: $\int_{a}^{b} x f(x) d x$
Let $\mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$
as, f(x) = f(a + b - x)
$\Rightarrow \mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}}(\mathrm{a}+\mathrm{b}-\mathrm{x}) \mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x}) \mathrm{d} \mathrm{x}$
$\Rightarrow \mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}}(\mathrm{a}+\mathrm{b}-\mathrm{x}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$
$\Rightarrow \mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}}(\mathrm{a}+\mathrm{b}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}-\int_{\mathrm{a}}^{\mathrm{b}}(\mathrm{x}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$
$\Rightarrow \mathrm{I}=\int_{2}^{\mathrm{b}}(\mathrm{a}+\mathrm{b}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}-\mathrm{I}$
$\Rightarrow 2 \mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}}(\mathrm{a}+\mathrm{b}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$
$\Rightarrow \mathrm{I}=\frac{(\mathrm{a}+\mathrm{b})}{2} \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$

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