Question

If the integral $I=\int e^{x^2}{x}^{3}dx$ $=e^{x^2}f\left(x\right)+c$ , 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, $x^2=t\Rightarrow 2xdx=dt$
$\therefore I=\int e^t\cdot \frac{t}{2}dt$
Using integration by parts, we get,
$I=\frac{1}{2}\left(t{e}^t-\int 1\cdot e^{t}dt\right)$
$=\frac{1}{2}\left(t{e}^t-e^t\right)+c$
$=e^{x^2}\left(\frac{x^2-1}{2}\right)+c$
$\therefore f\left(x\right)=\frac{x^2-1}{2}\Rightarrow f\left(2\right)=\frac{3}{2}$