Let f, g : (0, ∞) → R be two functions defined by Then the value of is equal to.

Engineering

Mathematics

Properties of Definite Integral

Question

Let f, g : (0, ∞) → R be two functions defined by $f (x) = \int_{- x}^{x} (| t | - t^{2}) e^{- t^{2}} dt and g (x) = \int_{0}^{x^{2}} t^{\frac{1}{2}} e^{- t^{2}} dt$ Then the value of $9 (f (\sqrt{\log_{e} 9}) + g (\sqrt{\log_{e} 9}))$ is equal to.

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g (x) = \int_{0}^{x^{2}} \sqrt{t} e^{- t} dt

Let t = y²

dt = 2y dy

g (x) = \int_{0}^{x} 2 y^{2} e^{- y^{2}} dy = 2 \int_{0}^{x} t^{2} e^{- t^{2}} dt

f (x) = 2 \int_{0}^{x} (t - t^{2} e^{- t^{2}} dt

Nowf (x) + g (x) = 2 \int_{0}^{x} {te}^{- t^{2}} dt

Let t² = p

2tdt = dp

f (x) + g (x) = \int_{0}^{x^{2}} {\bar{e}}^{p} dp = - {(e^{- p})}_{0}^{x^{2}} = 1 - e^{- x^{2}}

9 (f (\sqrt{\log_{e} 9}) + g (\sqrt{\log_{e} 9})) = 9 (1 - {\bar{e}}^{ℓn 9}) = 9 (1 - \frac{1}{9}) = 8