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

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Answer 1 · 2014-09-26T09:13:10

Begin by making the #u#-substitution.

Let #u=sqrt(x)#

#u=x^(1/2)#

#du= 1/2*x^((-1)/2) dx#

#du= 1/(2*x^((1)/2)) dx#

#2du= (dx)/(x^((1)/2))#

#2du= (dx)/sqrt(x) =># Notice that is can be substituted in the integral

#int(cos sqrt(x))/(sqrt(x))dx=int cos(u)*2du=2intcos(u)du#

#=2[sin(u)]+C#

Rewrite in terms of #x#

#=2[sin(sqrt(x))]+C#