If f ( x ) = ∫ x a t 3 e t d t , t h e n d d x f ( x ) = [MP PET 1989] Options: A . &nbspex(x3+3x2) B . &nbspx3ex C . &nbspa3ex D . &nbspNone of these

Answer: Option B : B f ( x ) = ∫ x a t 3 e t d t = ∫ 0 a t 3 . e t d t + ∫ x 0 t 3 e t d t ⇒ d f ( x ) d x = d d x ( ∫ 0 a t 3 . e t d t ) + d d x ( ∫ x 0 t 3 . e t d t ) = x 3 e x

If f(x)=∫xa t3et dt, then ddxf(x)= [MP PET 1989]

Question

f (x) = \int_{a}^{x} t^{3} e^{t} d t, t h e n \frac{d}{d x} f (x) =

[MP PET 1989]

Options:

A . ex(x3+3x2)

B . x3ex

C . a3ex

D . None of these

Answer: Option B
:
B

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) = x^{3} e^{x}

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