Solve the Following Questions

Question:

If $0

to :

  1. $x\left(\frac{1+x}{1-x}\right)+\log _{e}(1-x)$

  2. $x\left(\frac{1-x}{1+x}\right)+\log _{e}(1-x)$

  3. $\frac{1-x}{1+x}+\log _{e}(1-x)$

  4. $\frac{1+x}{1-x}+\log _{e}(1-x)$


Correct Option: 1

Solution:

Let $\mathrm{t}=\frac{3}{2} \mathrm{x}^{2}+\frac{5}{3} \mathrm{x}^{3}+\frac{7}{4} \mathrm{x}^{4}+\ldots . \infty$

$=\left(2-\frac{1}{2}\right) x^{2}+\left(2-\frac{1}{3}\right) x^{3}+\left(2-\frac{1}{4}\right) x^{4}$

$+\ldots . \infty$

$=2\left(x^{2}+x^{3}+x^{4}+\ldots \infty\right)-\left(\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+\ldots \infty\right)$

$=\frac{2 x^{2}}{1-x}-(\ell n(1-x)-x)$

$\Rightarrow t=\frac{2 x^{2}}{1-x}+x-\ell n(1-x)$

$\Rightarrow \mathrm{t}=\frac{\mathrm{x}(1+\mathrm{x})}{1-\mathrm{x}}-\ell \mathrm{n}(1-\mathrm{x})$

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