July 22nd, 2014, 01:44 PM  #1 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,947 Thanks: 1139 Math Focus: Elementary mathematics and beyond  Integration Challenge
$\displaystyle \int\frac{dx}{\sec x+1}$

July 22nd, 2014, 04:25 PM  #2 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,203 Thanks: 901 Math Focus: Wibbly wobbly timeywimey stuff.  $\displaystyle \int \frac{1}{sec(x) + 1}~dx$ $\displaystyle = \int \frac{1}{sec(x)} \cdot \frac{sec(x)  1}{sec(x)  1}~dx$ $\displaystyle = \int \frac{sec(x)  1}{sec^2(x)  1}~dx$ $\displaystyle = \int \frac{sec(x)}{tan^2(x)}~dx  \int cot^2(x)~dx$ $\displaystyle = \int \frac{cos(x)}{sin^2(x)}~dx  \int cot^2(x)~dx$ Both of these integrations are reasonably trivial so I will skip it to the penultimate expression: $\displaystyle = (csc(x))  (cot(x)  x)$ $\displaystyle = csc(x) + cot(x) + x + C$ Dan Do we have spoiler tags here? Last edited by topsquark; July 22nd, 2014 at 04:29 PM. 
July 22nd, 2014, 04:48 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, greg1313! Quote:
Multiply by $\frac{\sec x  1}{\sec x  1}\!:$ $\dfrac{1}{\sec x + 1}\cdot \dfrac{\sec x  1}{\sec x  1} \:=\:\dfrac{\sec x1}{\sec^2\!x1} \;=\;\dfrac{\sec x 1}{\tan^2\!x} \;=\;\dfrac{\sec x}{\tan^2\!x}  \dfrac{1}{\tan^2\!x}$ $\qquad =\;\csc x\cot x  \cot^2\!x \;=\; \csc x\cot x  (\csc^2\!x1)$ We have: $\displaystyle \:\int(\csc x\cot x  \csc^2\!x + 1)\,dx \;=\;\csc x + \cot x + x + C$  
July 22nd, 2014, 05:45 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,203 Thanks: 901 Math Focus: Wibbly wobbly timeywimey stuff. 
is there an echo in here? Dan 
July 22nd, 2014, 08:22 PM  #5 
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, Dan! Sorry for the echo. I was so proud of my solution that I didn't see your post. 
July 22nd, 2014, 09:44 PM  #6 
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  
July 23rd, 2014, 11:04 AM  #7 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,203 Thanks: 901 Math Focus: Wibbly wobbly timeywimey stuff.  

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