Thursday, December 30, 2021

12 Days of Christmas Integrals: ∫ cos (√x) dx

12 Days of Christmas Integrals:  ∫ cos (√x) dx


On the Sixth day of Christmas Integrals, the integral featured today is...


∫ cos (√x) dx


Handling this integral will require two integral methods.  First substitution:


Let u = √x = x^(1/2)

Then:

du = 1/2 ∙ x^(-1/2) dx

2 ∙ x^(1/2) du = dx

2 ∙ u du = dx


∫ cos(√x) dx


= ∫ cos(x^(1/2)) dx


= ∫ 2 ∙ u ∙ cos(u) du


At this point, we now apply Integration by Parts:


w = 2 ∙ u

dw = 2 du


dv = cos(u) du

v = sin(u)


= 2 ∙ u ∙ sin(u) - ∫ 2 ∙ sin(u) du


= 2 ∙ u ∙ sin(u) +  2 ∙ cos(u) + C


Recall u = x^(1/2):


= 2 ∙ x^(1/2) ∙ sin(x^(1/2)) + 2 ∙ cos(x^(1/2)) + C


Eddie 


All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


1 comment:

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