If f x - a x t 3 e t dt- then the value of df x - dxA- exx3-3 x2B- x3 exC- e 3 e aD- None of these

The correct option is B x^3e^xf(x) = ∫_a^x t^3e^t dt = ∫_a^0 t^3 .e^t dt + ∫_0^x t^3 e^t dt ⇒df(x)dx = ddx(∫_a^0 t^3.e^t dt ) + ddx(∫_0^x t^3.e^t dt ) ⇒df( ...

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Byju's Answer

Standard XII

Mathematics

Higher Order Derivatives

If f x =∫ a x...

Question

If $f (x) = \int_{a}^{x} t^{3} e^{t} d t$ , then the value of $\frac{d f (x)}{d x}$ is

e^{x} (x^{3} + 3 x^{2})

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x^{3} e^{x}

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e^{3} e^{a}

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None of these

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Solution

The correct option is B $x^{3} e^{x}$
$f (x) = \int_{a}^{x} t^{3} e^{t} d t = \int_{a}^{0} t^{3} . e^{t} d t + \int_{0}^{x} t^{3} e^{t} d t \Rightarrow \frac{d f (x)}{d x} = \frac{d}{d x} (\int_{a}^{0} t^{3} . e^{t} d t) + \frac{d}{d x} (\int_{0}^{x} t^{3} . e^{t} d t) \Rightarrow \frac{d f (x)}{d x} = 0 + x^{3} e^{x} = x^{3} e^{x}$

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Similar questions

If $f (x) = \int_{a}^{x} t^{3} e^{t} d t, t h e n \frac{d}{d x} f (x) =$ [MP PET 1989]

$\int e^{(x + \frac{1}{x})} (1 + x - \frac{1}{x}) d x = e^{(x + \frac{1}{x})} f (x) + c t h e n \frac{d f (x)}{d x}$

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Q. If g(x) = 2f(x) + x + log (f(x)) then

\frac{d (g (x))}{d (f (x))}

_________

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is
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(b) −1
(c) 1/2
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If f(x)=∫xat3etdt, then the value of df(x)dx is

The correct option is B x3exf(x)=∫xat3etdt=∫0at3.etdt+∫x0t3etdt⇒df(x)dx=ddx(∫0at3.etdt)+ddx(∫x0t3.etdt)⇒df(x)dx=0+x3ex=x3ex

If $f (x) = \int_{a}^{x} t^{3} e^{t} d t$ , then the value of $\frac{d f (x)}{d x}$ is