_ ^ e^x dx a + b e^x =

MCQ

$\int_{}^{} {\frac{{{e^x}dx}}{{\sqrt {a + b{e^x}} }}} = $

✓
$2/b\sqrt {a + b{e^x}} + c$
B
$2b\sqrt {a + b{e^x}} + c$
C
$\frac{1}{{2b}}\sqrt {a + b{e^x}} $
D
$\frac{a}{b}\sqrt {a + b{e^x}} + c$

✓

Answer

Correct option: A.

$2/b\sqrt {a + b{e^x}} + c$

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

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