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March 8th, 2018, 04:55 PM  #1 
Newbie Joined: Mar 2018 From: Houston,TX Posts: 2 Thanks: 0  May someone please help me with this question?
Hello, I would like to know if you can help me with this problem the intergal of (cos (6x)+1), I’ve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.

March 8th, 2018, 06:03 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra 
\begin{align*}\cos{(2A)} &= \cos^2{(A)}  \sin^2{(A)} \\ &= \cos^2{(A)}  \big(1  \cos^2{(A)}\big) \\ &= 2\cos^2{(A)}  1 \\ \cos{(2A)} + 1 &= 2\cos^2{(A)} \\[8pt] \sqrt{\cos{(2A)} + 1} &= \sqrt{2\cos^2{(A)}} \\ &= \sqrt{2\vphantom{\cos^2{(A)}}}\sqrt{\cos^2{(A)}} \\ &= \sqrt{2}\cos{(A)}\end{align*} Last edited by v8archie; March 8th, 2018 at 06:05 PM. 
March 8th, 2018, 06:17 PM  #3 
Newbie Joined: Mar 2018 From: Houston,TX Posts: 2 Thanks: 0 
Oh my goodness, thank you so much for your help, I greatly appreciate it. I’ve been trying to figure this out for so long. I have and solution, but is this the same as this square root of 2, I understand where the 1/3 (4 square root of 2 comes from).

March 8th, 2018, 07:53 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra 
The $\sqrt2$ comes from the derivation above. It gets multiplied by the $1$ in the following. \begin{align*}\sin{(3x)} &= \sin{(2x)}\cos{(x)} + \cos{(2x)}\sin{(x)} \\ &= \big(2\sin{(x)}\cos{(x)}\big)\cos{(x)} + \big(2\cos^2{(x)}1\big)\sin{(x)} \\ &= 4\sin{(x)}\cos^2{(x)}  \sin{(x)} \\ &= \big(4\cos^2{(x)}  1\big)\sin{(x)} \end{align*} I don't know why Symbolab decided to write the answer in trigonometric functions of $x$. I'd have left it in terms of trigonometric functions of $3x$. Last edited by v8archie; March 8th, 2018 at 07:55 PM. 

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