Question

If the integral $Ι=\displaystyle \int e^{x^{2}}x^{3}dx$ $=e^{x^{2}}f\left(x\right)+c$ , where $c$ is the constant of integration and $f\left(1\right)=0,$ then the value of $f\left(2\right)$ is equal to

• A. $4$
• B. $\frac{5}{2}$
• C. $\frac{3}{2}$
• D. $3$

Answer 1

Answer: (C)

Let, $x^{2}=t\Rightarrow 2xdx=dt$
$\therefore Ι=\displaystyle \int e^{t} \cdot \frac{t}{2} d t$
Using integration by parts, we get,
$Ι=\frac{1}{2}\left(t e^{t} - \displaystyle \int 1 \cdot e^{t} d t\right)$
$=\frac{1}{2}\left(t e^{t} - e^{t}\right)+c$
$=e^{x^{2}}\left(\frac{x^{2} - 1}{2}\right)+c$
$\therefore f\left(x\right)=\frac{x^{2} - 1}{2}\Rightarrow f\left(2\right)=\frac{3}{2}$