Question

If $\int \frac{x^2-x+1}{(x^2+1{)}^{3/2}}{e}^{x}dx=e^{x}f\left(x\right)+C,$ where $C$ is an integration constant, then

• A. $f\left(x\right)$ is an even function
• B. $f\left(x\right)$ is a bounded function
• C. $f\left(x\right)$ has two points of extrema
• D. Range of $f\left(x\right)$ is $(0,1]$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $f\left(x\right)$ is an even function
B $f\left(x\right)$ is a bounded function
D Range of $f\left(x\right)$ is $(0,1]$

$I=\int e^x[\frac{1}{(x^2+1{)}^{1/2}}+\frac{-x}{(x^2+1{)}^{3/2}}]dx=\int e^x\left[g\right(x)+g^{\prime }(x\left)\right]dx\Rightarrow I=e^{x}g\left(x\right)+C\Rightarrow f\left(x\right)=g\left(x\right)=\sqrt{\frac{1}{1+x^2}}$

So, $f\left(x\right)$ is an even function.
The minimum value of $1+x^2$ is $1$
So the maximum value of $f\left(x\right)$ is $1$
So the range of $f\left(x\right)$ is $(0,1]$