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March 2nd, 2017, 11:38 AM  #1 
Newbie Joined: Mar 2017 From: Europe Posts: 2 Thanks: 0  Constant of the Integral as a derivative?
Hey, I was solving this integral on a website: ∫ sin(x)cos(x)dx And I got two answers. One of them was this one: ½sin^(2)x + c and the other one was this one: ½cos^(2)x + c But on the site it shows this solution: ½cos^(2)x + c' where the constant is a derivative. I plugged the integral on the integral calculator website and the constant is not a derivative. Does anybody know the thought process behind writing the constant as a derivative? P.S: In case you're wondering, here are the steps for the second solution (copypaste): Now substitute into the formula: ∫ uvdx = u ∫vdx − ∫u'(∫vdx) dx This gives: ∫ sin(x)cos(x)dx = cos(x) × cos(x) − ∫ sin(x) × cos(x)dx = cos2x − ∫ sin(x)cos(x)dx Again we have the same integral on both sides (except one is subtracted) ... ... so bring the right hand one over to the left and we get: 2∫ sin(x)cos(x)dx = cos2x ⇒ ∫ sin(x)cos(x)dx = ½cos2x + c' 
March 2nd, 2017, 11:47 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,208 Thanks: 516 Math Focus: Calculus/ODEs 
My guess would be that the prime notation being used on the constant of integration does not denote differentiation, but rather distinguishes it from the original constant after dividing through by 2.

March 2nd, 2017, 12:20 PM  #3 
Senior Member Joined: May 2016 From: USA Posts: 1,207 Thanks: 494 
$\displaystyle \int \sin(x) * \cos(x)\ dx = 0.5 * \sin^2(x) + C_1 =$ $0.5\{1  \cos^2(x)\} + C_1 = \ 0.5 * \cos^2(x) + 0.5 + C_1 = \ 0.5 * \cos^2(x) + C_2.$ Or this may be what was intended. The constant is not the same when the integral is expressed in terms of the sine as it is when the integral is expressed in terms of the cosine. Using the prime sign is sloppy notation. Last edited by skipjack; March 2nd, 2017 at 05:38 PM. 
March 2nd, 2017, 12:23 PM  #4 
Newbie Joined: Mar 2017 From: Europe Posts: 2 Thanks: 0 
That makes sense. Thanks for clearing this out!


Tags 
constant, derivative, integral, trigonometry 
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