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Differentiation under Integral Sign

If ∫ x2· e-2 ...

Question

If $\int x^{2} \cdot e^{- 2 x} d x = e^{- 2 x} (a x^{2} + b x + c) + d,$ where $d$ is the constant of integration, then the value of $∣ ∣ ∣ \frac{a}{b c} ∣ ∣ ∣$ is

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Solution

$\int x^{2} \cdot e^{- 2 x} d x = e^{- 2 x} (a x^{2} + b x + c) + d,$

Differentiating both the sides
$x^{2} \cdot e^{- 2 x} = e^{- 2 x} (2 a x + b - 2 (a x^{2} + b x + c)) \Rightarrow x^{2} = - 2 a x^{2} + (2 a - 2 b) x + b - 2 c \Rightarrow - 2 a = 1, 2 a - 2 b = 0, b - 2 c = 0 \Rightarrow a = b = - \frac{1}{2}, c = - \frac{1}{4} \Rightarrow ∣ ∣ ∣ \frac{a}{b c} ∣ ∣ ∣ = 4$

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