MCQ
The value of $\int_{-1}^{1} \frac{(1+\sqrt{|x|-x}) e^{x}+(\sqrt{|x|-x}) e^{-x}}{e^{x}+e^{-x}} d x$ is equal to
- A$3-\frac{2 \sqrt{2}}{3}$
- B$2+\frac{2 \sqrt{2}}{3}$
- C$1-\frac{2 \sqrt{2}}{3}$
- ✓$1+\frac{2 \sqrt{2}}{3}$
✓
Answer
Correct option: D.
$1+\frac{2 \sqrt{2}}{3}$
(D) $1+\frac{2 \sqrt{2}}{3}$
$I=\int_{-1}^{1} \frac{(1+\sqrt{|-x|-(-x)}) \mathrm{e}^{-\mathrm{x}}+(\sqrt{|-\mathrm{x}|-(-\mathrm{x})}) \mathrm{e}^{-(-\mathrm{x})}}{\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{-(-\mathrm{x})}} \mathrm{dx}$
$\Rightarrow \mathrm{I}=\int_{-1}^{1} \frac{(1+\sqrt{|\mathrm{x}|+\mathrm{x}}) \mathrm{e}^{-\mathrm{x}}+(\sqrt{|\mathrm{x}|+\mathrm{x}}) \mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{\mathrm{x}}} \mathrm{dx}$
$\Rightarrow 2 \mathrm{I}=\int_{-1}^{1} \frac{(1+\sqrt{|\mathrm{x}|+\mathrm{x}}+\sqrt{|\mathrm{x}|-\mathrm{x}})\left(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}\right)}{\left(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}\right)} \mathrm{dx}$
$\Rightarrow 2 I=\int_{-1}^{1}(1+\sqrt{|x|+x}+\sqrt{|x|-x}) d x$
$\Rightarrow 2 \mathrm{I}=2 \int_{0}^{1}(1+\sqrt{|\mathrm{x}|+\mathrm{x}}+\sqrt{|\mathrm{x}|-\mathrm{x}}) \mathrm{dx}$
$\Rightarrow 2 \mathrm{I}=2 \int_{0}^{1}(1+\sqrt{2 \mathrm{x}}+\sqrt{0}) \mathrm{dx}$
$\Rightarrow \mathrm{I}=\int_{0}^{1}(1+\sqrt{2 \mathrm{x}}) \mathrm{dx}=\left[\mathrm{x}+\frac{2 \sqrt{2}}{3} \mathrm{x}^{3 / 2}\right]_{0}^{1}$
$\Rightarrow \mathrm{I}=\frac{2 \sqrt{2}}{3}+1$
$I=\int_{-1}^{1} \frac{(1+\sqrt{|-x|-(-x)}) \mathrm{e}^{-\mathrm{x}}+(\sqrt{|-\mathrm{x}|-(-\mathrm{x})}) \mathrm{e}^{-(-\mathrm{x})}}{\mathrm{e}^{-\mathrm{x}}+\mathrm{e}^{-(-\mathrm{x})}} \mathrm{dx}$
$\Rightarrow \mathrm{I}=\int_{-1}^{1} \frac{(1+\sqrt{|\mathrm{x}|+\mathrm{x}}) \mathrm{e}^{-\mathrm{x}}+(\sqrt{|\mathrm{x}|+\mathrm{x}}) \mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{\mathrm{x}}} \mathrm{dx}$
$\Rightarrow 2 \mathrm{I}=\int_{-1}^{1} \frac{(1+\sqrt{|\mathrm{x}|+\mathrm{x}}+\sqrt{|\mathrm{x}|-\mathrm{x}})\left(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}\right)}{\left(\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}\right)} \mathrm{dx}$
$\Rightarrow 2 I=\int_{-1}^{1}(1+\sqrt{|x|+x}+\sqrt{|x|-x}) d x$
$\Rightarrow 2 \mathrm{I}=2 \int_{0}^{1}(1+\sqrt{|\mathrm{x}|+\mathrm{x}}+\sqrt{|\mathrm{x}|-\mathrm{x}}) \mathrm{dx}$
$\Rightarrow 2 \mathrm{I}=2 \int_{0}^{1}(1+\sqrt{2 \mathrm{x}}+\sqrt{0}) \mathrm{dx}$
$\Rightarrow \mathrm{I}=\int_{0}^{1}(1+\sqrt{2 \mathrm{x}}) \mathrm{dx}=\left[\mathrm{x}+\frac{2 \sqrt{2}}{3} \mathrm{x}^{3 / 2}\right]_{0}^{1}$
$\Rightarrow \mathrm{I}=\frac{2 \sqrt{2}}{3}+1$
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