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7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
MCQ
The integral $\int {\frac{{xdx}}{{2 - {x^2} + \sqrt {2 - {x^2}} }}} $ equals
  • A
    $ \log \left| {1 + \sqrt {2 + {x^2}} } \right| + c$
  • ✓
    $ - \log \left| {1 + \sqrt {2 - {x^2}} } \right| + c$
  • C
    $ - x\log \left| {1 - \sqrt {2 - {x^2}} } \right| + c$
  • D
    $ x\log \left| {1 - \sqrt {2 + {x^2}} } \right| + c$
✓

Answer

Correct option: B.
$ - \log \left| {1 + \sqrt {2 - {x^2}} } \right| + c$
b
$\mathrm{I}=\int \frac{x d x}{2-x^{2}+\sqrt{2-x^{2}}}$

Put $t=\sqrt{2-x^{2}}, \frac{d t}{d x}$

$=\frac{1}{2 \sqrt{2-x^{2}}} \cdot(-2 x)$

$\Rightarrow-t d t=x d x$

$\therefore $

$I=\int \frac{(-t) d t}{t^{2}+t}$

$=-\int \frac{1}{t+1} d t$

$=-\log |t+1|$

$=-\log |\sqrt{2-x^{2}}+1|+c$

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