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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
MCQ
If $\int \limits_0^1 \frac{1}{\left(5+2 x -2 x ^2\right)\left(1+ e ^{(2-4 x)}\right)} dx =\frac{1}{\alpha} \log _{ e }\left(\frac{\alpha+1}{\beta}\right)$ $\alpha, \beta > 0$, then $\alpha^4-\beta^4$ is equal to$:$
  • A
    $21$
  • B
    $0$
  • C
    $19$
  • D
    $-21$
✓

Answer

$I=\int \limits_0^1 \frac{d x}{\left(5+2 x-2 x^2\right)\left(1+ e ^{2-4 \pi}\right)}$
$x \rightarrow 1-x$
$I=\int \limits_0^1 \frac{e^{2-4 x} d x}{\left(5+2 x-2 x^2\right)\left(1+ e ^{2-4 x}\right)}$
Add $(i)$ and $(ii)$
$2 I=\int \limits_0^1 \frac{d x}{5+2 x-2 x^2}$
$=\int \limits_0^1 \frac{d x}{2\left(\frac{11}{4}-\left(x-\frac{1}{2}\right)^2\right)}$
$I=\frac{1}{\sqrt{11}} \ln \left(\frac{\sqrt{11}+1}{\sqrt{10}}\right)$
$\alpha=\sqrt{11}$
$\beta=\sqrt{10}$
$\alpha^4-\beta^4$
$=121-100$
$=21$

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