How do I solve this equation?

Question

#d/dx int_0^x cos(2piu)du#

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Answer 1 · 2018-04-15T19:51:02

#= cos (2 \ pi \ x )#

Liebnitz Rule for differentiating under the integral, here in expurgated form:

#d/(dx) int_(alpha(x))^(beta(x))\ f (u) \ d u = f(beta(x)) (d beta)/(dx) - f(alpha(x)) (d alpha)/(dx)#

Here that means:

#d/(dx) int_(0)^(x) cos (2 pi u) \ d u #

#= cos (2 pi x) * 1 - cos ( 2 pi 0 ) * 0 #

#= cos (2 \ pi \ x )#