Question

The integral $Ι=\displaystyle \int e^{x}\left(\frac{1 + sin x}{1 + cos ⁡ x}\right)dx=e^{x}f\left(x\right)+C$ (where, $C$ is the constant of integration). Then, the range of $y=f\left(x\right)$ (for all $x$ in the domain of $f\left(x\right)$ ) is

• A. $\left[- 1,1\right]$
• B. $\left(- \in fty , \in fty\right)$
• C. $\left(- 1,1\right)$
• D. $[0 , \infty)$

Answer 1

Answer: (B)

The given integral $I =\int e^{x}\left(\frac{1+2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}\right) d x$
$=\int e^{x}\left(\frac{1}{2} \sec ^{2}\left(\frac{x}{2}\right)+\tan \left(\frac{x}{2}\right)\right) d x$
$=e^{x} \cdot \tan \left(\frac{x}{2}\right)+C\left\{\because \int e^{x}\left(f(x)+f^{\prime}(x)\right)\left(d x=e^{x} f(x)+C\right)\right\}$
Thus, $f(x)=\tan \left(\frac{x}{2}\right)$ whose range is $(-\infty, \infty)$