# How do you integrate #(sin(2x))/(5-sin(x))^(1/2) dx#?

If you so choose, you can factorize.

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To integrate ( \frac{\sin(2x)}{\sqrt{5 - \sin(x)}} , dx ), you can use the substitution method. Let ( u = 5 - \sin(x) ). Then ( du = -\cos(x) , dx ).

Rewrite the integral in terms of ( u ):

[ \begin{aligned} \int \frac{\sin(2x)}{\sqrt{5 - \sin(x)}} , dx &= -2 \int \frac{1}{\sqrt{u}} , du \ &= -4 \sqrt{u} + C \end{aligned} ]

Substitute back for ( u ):

[ -4 \sqrt{5 - \sin(x)} + C ]

Where ( C ) is the constant of integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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