_ ^ 2x a^2 + b^2 ^2 x ;dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{\sin 2x}}{{{a^2} + {b^2}{{\sin }^2}x}}} \;dx = $

✓
$\frac{1}{{{b^2}}}\log ({a^2} + {b^2}{\sin ^2}x) + c$
B
$\frac{1}{b}\log ({a^2} + {b^2}{\sin ^2}x) + c$
C
$\log ({a^2} + {b^2}{\sin ^2}x) + c$
D
${b^2}\log ({a^2} + {b^2}{\sin ^2}x) + c$

✓

Answer

Correct option: A.

$\frac{1}{{{b^2}}}\log ({a^2} + {b^2}{\sin ^2}x) + c$

a
(a) Put ${a^2} + {b^2}{\sin ^2}x = t \Rightarrow {b^2}\sin 2x\,dx = dt,$ then
$\int_{}^{} {\frac{{\sin 2x}}{{{a^2} + {b^2}{{\sin }^2}x}}\,dx = \frac{1}{{{b^2}}}\int_{}^{} {\frac{{dt}}{t} = \frac{1}{{{b^2}}}\log t + c} } $
$ = \frac{1}{{{b^2}}}\log ({a^2} + {b^2}{\sin ^2}x) + c.$

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