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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_{-1 / 2}^{1 / 2}(\cos x)\left[\log \left(\frac{1-x}{1+x}\right)\right] d x=$
  • ✓
    $0$
  • B
    1
  • C
    $e^{\frac{1}{2}}$
  • D
    $2 e ^{\frac{1}{2}}$
✓

Answer

Correct option: A.
$0$
(A)
Let $f(x)=\cos x \log \left(\frac{1-x}{1+x}\right)$
$\therefore \quad f(-x)=\cos x \log \left(\frac{1-x}{1+x}\right)^{-1}$
$=-\cos x \log \left(\frac{1-x}{1+x}\right)=-f(x)$
$\begin{array}{ll}\therefore & f(x) \text { is an odd function. } \\ \therefore & \int_{-\frac{1}{2}}^{\frac{1}{2}} \cos x \log \left(\frac{1-x}{1+x}\right) d x=0\end{array}$

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