{"id":220,"date":"2019-06-26T23:04:37","date_gmt":"2019-06-26T23:04:37","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=220"},"modified":"2019-12-09T22:14:36","modified_gmt":"2019-12-09T22:14:36","slug":"one-equals-zero-integral-form","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/one-equals-zero-integral-form\/","title":{"rendered":"One Equals Zero: Integral Form"},"content":{"rendered":"\n<p>Consider the following\u00a0integral:<\/p>\n\n\n\n<p style=\"text-align:center\">INTEGRAL (1\/x) dx<\/p>\n\n\n\n<p>Perform integration by parts: let<\/p>\n\n\n\n<p style=\"text-align:center\">u = 1\/x , dv = dx\u00a0<br>du = -1\/x<sup>2<\/sup>\u00a0dx , v = x<\/p>\n\n\n\n<p>Then obtain:<\/p>\n\n\n\n<p style=\"text-align:center\">INTEGRAL (1\/x) dx = (1\/x)*x &#8211; INTEGRAL x (-1\/x<sup>2<\/sup>) dx\u00a0<br>= 1 + INTEGRAL (1\/x) dx<\/p>\n\n\n\n<p>which implies that 0 = 1.<\/p>\n\n\n\n<p>What&#8217;s wrong with this calculation?<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>This is common mistake using integration by parts in\u00a0calculus. Students often forget about the constant of integration for indefinite integrals. In this case, the constants on both sides will differ by 1.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>\u00a0<br>Su, Francis E., et al. &#8220;One Equals Zero: Integral Form.&#8221;\u00a0<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n<p><strong>References:<\/strong><br>R. Vakil, <em><a href=\"http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1895997046\/ref=nosim\/mathfunfacts-20\">A Mathematical Mosaic<\/a><\/em>, 1996. p. 199.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:   <\/strong><br>Joshua Sabloff <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the following\u00a0integral: INTEGRAL (1\/x) dx Perform integration by parts: let u = 1\/x , dv = dx\u00a0du = -1\/x2\u00a0dx&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[2,17,3,12,18],"class_list":["post-220","page","type-page","status-publish","hentry","tag-algebra","tag-algebra-false-proof","tag-easy","tag-other","tag-paradox"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/220","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=220"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/220\/revisions"}],"predecessor-version":[{"id":1550,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/220\/revisions\/1550"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=220"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}