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Answer 1 · 2017-04-07T17:44:34

$\int (\frac{cos (2 x)}{\sqrt{1 + sin (2 x)}}) d x = \sqrt{1 + sin (2 x)} + C$ $int(cos(2x)/sqrt(1+sin(2x)))dx = sqrt(1+sin(2x))+ C$

Let $u = 1 + sin (2 x)$ $u = 1 + sin(2x)$ , then $d u = 2 cos (2 x) d x$ $du = 2cos(2x)dx$

$cos (2 x) d x = \frac{1}{2} d u$ $cos(2x)dx = 1/2du$

$\int (\frac{cos (2 x)}{\sqrt{1 + sin (2 x)}}) d x =$ $int(cos(2x)/sqrt(1+sin(2x)))dx =$

$\frac{1}{2} \int (\frac{1}{\sqrt{u}}) d u =$ $1/2int(1/sqrtu)du =$

$\sqrt{u} + C =$ $sqrtu+C =$

$\sqrt{1 + sin (2 x)} + C$ $sqrt(1+sin(2x))+ C$