Question

If $f\left(x\right)={\int }_a^x{t}^3{e}^{t}dt,then\frac{d}{dx}f\left(x\right)=$ [MP PET 1989]

• A. $e^x(x^3+3x^2)$
• B. $x^3{e}^x$
• C. $a^3{e}^x$
• D. $Noneofthese$

Answer 1

The correct option is B $x^3{e}^x$

$f\left(x\right)={\int }_a^x{t}^3{e}^{t}dt={\int }_a^0{t}^3.e^{t}dt+{\int }_0^x{t}^3{e}^{t}dt$
$\Rightarrow \frac{df\left(x\right)}{dx}=\frac{d}{dx}\left({\int }_a^0{t}^3.e^{t}dt\right)+\frac{d}{dx}\left({\int }_0^x{t}^3.e^{t}dt\right)=x^3{e}^x$