Question 344596
Apply the Leibniz rule (see http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign#Higher_dimensions)
Follow the Example 6 discussion.
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{{{ int( cos(u^2+1), du, a(y), b(y) )=cos(b^2+1)*(db/dy)-cos(a^2+1)*(da/dy)}}}
The third term is not present since 
d/dx(cos(u^2+1))=0
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{{{a(y)=sin(y)}}}
{{{da/dy=cos(y)}}}
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{{{b(y)=sqrt(y)}}}
{{{db/dy=1/(2*sqrt(y))}}}
Substituting,
{{{ int( cos(u^2+1), du, sin(y), sqrt(y) )=(cos(y+1))/(2*sqrt(y))-cos(sin^2(y)+1)*cos(y)}}}