# How do you use Integration by Substitution to find #intcos(sqrt(x))/sqrt(x)dx#?

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To integrate the given expression, we'll use the substitution method. Let ( u = \sqrt{x} ). Then, ( du = \frac{1}{2\sqrt{x}} dx ).

Now, we can rewrite the integral in terms of ( u ):

[ \int \frac{\cos(\sqrt{x})}{\sqrt{x}} dx = \int \frac{\cos(u)}{u} \cdot 2u , du ]

[ = 2\int \cos(u) , du ]

[ = 2\sin(u) + C ]

Substituting back for ( u ) gives:

[ = 2\sin(\sqrt{x}) + C ]

where ( C ) is the constant of integration.

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