Question

Integrate cosx/(sinx+√(sinx)
$sinx$
​

Answer 1

Given:

$\dfrac{cosx}{sinx+\sqrt{sinx} }$

To Find:

Integrate -   $\dfrac{cosx}{sinx+\sqrt{sinx} }$ .

Solution:

Let,

I = $\int{\frac{cos x}{sin x + \sqrt{sin x} } } \, dx$

now, consider that -

$sin x=t\\(cosx) dx= dt$

therefore, I become,

$I=\int {\dfrac{1}{t+\sqrt{t} } } \, dt\\ =\dfrac{log(t+\sqrt{t} )}{1+\dfrac{t^{3/2} }{3/2} } \\ =\dfrac{log(t+\sqrt{t} )}{\dfrac{3+2t^{3/2} }{3} }\\ =\dfrac{3log(t+\sqrt{t} )}{3+2t^{3/2} }$

Since,$sinx=t$

so, result is -

$I=\dfrac{3log(sinx+\sqrt{sinx} )\\}{3+2sinx^{3/2} }$