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प्रश्न
Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`
उत्तर
Let I = `int_(−9)^9 x^3/(4 − x^2).dx`
We know that, If f(−x) = f(x), f(x) is an even function. If f(−x) = −f(x), f(x) is an odd function.
f(x) = `x^3/(4 – x^2)`
∴ f(– x) = `(– x)^3/[4 – ( – x)^2]`
∴ f(– x) = `(−x^3)/(4 – x^2)`
∴ f(– x) = – f(x)
∴ If f(−x) = −f(x), f(x) is an odd function.
∴ `int_(−9)^9 x^3/(4 − x^2).dx = 0 ...[int_(−"a")^"a" f(x) = 0, if f(x) "odd function"]`
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