{"id":2283,"date":"2010-02-17T18:34:27","date_gmt":"2010-02-17T17:34:27","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=2283"},"modified":"2010-02-17T18:34:27","modified_gmt":"2010-02-17T17:34:27","slug":"integrating-absx-with-mathematica","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=2283","title":{"rendered":"Integrating Abs(x) with Mathematica"},"content":{"rendered":"

Someone recently emailed me to say that they thought Mathematica sucked because it couldn’t integrate abs(x) where abs stands for absolute value.\u00a0 The result he was expecting was<\/p>\n

\"\int|x| \,dx = \frac{|x|x}{2} \"<\/p>\n

When you try to do this in Mathematica 7.0.1, it appears that it simply can’t do it. The command<\/p>\n

Integrate[Abs[x], x]<\/pre>\n
just returns the integral unevaluated
\n\"unevaluated
\nI’ve come across this issue before and many people assume that Mathematica is just stupid…after all it appears that it can’t even do an integral expected of a high school student.  Well, the issue is that Mathematica is not a high school student and it assumes that x is a complex variable.  For complex x, this indefinite integral doesn’t have a solution!<\/p>\n
So, let’s tell Mathematica that x is real<\/p>\n
Integrate[Abs[x], x, Assumptions :> Element[x, Reals]]<\/pre>\n
\"Evaluated
\nwhich is Mathematica’s way of saying that the answer is -x^2\/2 for x<=0 and x^2\/2 otherwise, i.e. when x>0.  It’s not quite in the form we were originally expecting but a moments thought should convince you that they are the same thing.<\/p>\n
Interestingly, it seems that Wolfram Alpha guesses that you probably mean real x since it just evaluates the integral of abs(x) directly<\/a>.  It does, however, give the result in yet another form: in terms of the signum function<\/a>, sgn(x):
\n\"Wolfram<\/p>\n
A couple of weeks ago I am pretty sure that Wolfram Alpha gave exactly the same result as Mathematica 7.0.1 so I wonder if they have quietly upgraded the back-end Kernel of Wolfram Alpha.\u00a0 Perhaps this is how Mathematica version 8 will evaluate this result?<\/p>\n","protected":false},"excerpt":{"rendered":"
Someone recently emailed me to say that they thought Mathematica sucked because it couldn’t integrate abs(x) where abs stands for absolute value.\u00a0 The result he was expecting was When you try to do this in Mathematica 7.0.1, it appears that it simply can’t do it. The command Integrate[Abs[x], x] just returns the integral unevaluated I’ve […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6,4,8,33],"tags":[],"class_list":["post-2283","post","type-post","status-publish","format-standard","hentry","category-general-math","category-math-software","category-mathematica","category-wolfram-alpha"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-AP","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/2283","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2283"}],"version-history":[{"count":24,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/2283\/revisions"}],"predecessor-version":[{"id":2347,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/2283\/revisions\/2347"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2283"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2283"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}