{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":59521,"title":"Integrate a power tower","description":"Write a function to compute this integral\r\n\r\nwhere . That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; 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That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = intPowerTower(a)\r\n  I = integral(x^x^x^x^x^x^x^x,0,a);\r\nend","test_suite":"%%\r\na = 0;\r\nI = intPowerTower(a);\r\nassert(abs(I)\u003c1e-6)\r\n\r\n%%\r\na = 1/100;\r\nI = intPowerTower(a);\r\nI_correct = 0.00975627404012066;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/20;\r\nI = intPowerTower(a);\r\nI_correct = 0.04621245261821598;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/10;\r\nI = intPowerTower(a);\r\nI_correct = 0.0886781687569094;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/5;\r\nI = intPowerTower(a);\r\nI_correct = 0.1685639964895788;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.2071658901263798;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/8;\r\nI = intPowerTower(a);\r\nI_correct = 0.30215124860335973;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/2;\r\nI = intPowerTower(a);\r\nI_correct = 0.3972053202401857;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 2/3;\r\nI = intPowerTower(a);\r\nI_correct = 0.5277402852630483;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.5959989560650945;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 5/6;\r\nI = intPowerTower(a);\r\nI_correct = 0.6671963910854818;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nI = intPowerTower(a);\r\nI_correct = 0.822467033424113;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = (rand+3)/4;\r\nI = intPowerTower(a);\r\nI_correct = polyval([0.3875275 -0.9886411 1.132527 0.1505356 0.1405179],a);\r\nassert(abs(I-I_correct)\u003c5e-6)\r\n\r\n%%\r\nfiletext = fileread('intPowerTower.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'find') || contains(filetext,'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-03T15:06:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-12-31T18:50:11.000Z","updated_at":"2026-01-28T06:58:04.000Z","published_at":"2023-12-31T18:50:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute this integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^a {(x^x)^{(x^x)^{(x^x)\\\\ldots}} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le a \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":59521,"title":"Integrate a power tower","description":"Write a function to compute this integral\r\n\r\nwhere . That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.783px 8px; transform-origin: 122.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute this integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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AnGgx7jdhPsh4iVErvgcu4lMytGj7v56uakInAJxxMhe0XTq5tTcDXV+3D7Fyp+vuWVtajvpOSHQFAKnQBzxwiL39PSVRMkR+jEHMJzAbjWJ95g01cCqjBBYA4FTIA62KlyL6PeA+GVCd9p36R2afRi7y/k99qNfCeBkpNXGGr1SeTaPwCkQB43A4IckLptcNbnLxK+8K2kkzrNw6zuH27jR+9Emd1TmUVKO0giBXSBwKsSxi8ZQJYXAXhAQceylpVRPIdAQAiKOhhpDVRECe0FAxLGXllI9hUBDCIg4GmoMVUUI7AUBEcdeWkr1FAINISDiaKgxVBUhsBcE/g+tfmSG+LdlUAAAAABJRU5ErkJggg==\" width=\"135\" height=\"44\" alt=\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\" style=\"width: 135px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.3px 8px; transform-origin: 226.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = intPowerTower(a)\r\n  I = integral(x^x^x^x^x^x^x^x,0,a);\r\nend","test_suite":"%%\r\na = 0;\r\nI = intPowerTower(a);\r\nassert(abs(I)\u003c1e-6)\r\n\r\n%%\r\na = 1/100;\r\nI = intPowerTower(a);\r\nI_correct = 0.00975627404012066;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/20;\r\nI = intPowerTower(a);\r\nI_correct = 0.04621245261821598;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/10;\r\nI = intPowerTower(a);\r\nI_correct = 0.0886781687569094;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/5;\r\nI = intPowerTower(a);\r\nI_correct = 0.1685639964895788;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.2071658901263798;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/8;\r\nI = intPowerTower(a);\r\nI_correct = 0.30215124860335973;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/2;\r\nI = intPowerTower(a);\r\nI_correct = 0.3972053202401857;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 2/3;\r\nI = intPowerTower(a);\r\nI_correct = 0.5277402852630483;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.5959989560650945;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 5/6;\r\nI = intPowerTower(a);\r\nI_correct = 0.6671963910854818;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nI = intPowerTower(a);\r\nI_correct = 0.822467033424113;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = (rand+3)/4;\r\nI = intPowerTower(a);\r\nI_correct = polyval([0.3875275 -0.9886411 1.132527 0.1505356 0.1405179],a);\r\nassert(abs(I-I_correct)\u003c5e-6)\r\n\r\n%%\r\nfiletext = fileread('intPowerTower.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'find') || contains(filetext,'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-03T15:06:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-12-31T18:50:11.000Z","updated_at":"2026-01-28T06:58:04.000Z","published_at":"2023-12-31T18:50:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute this integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^a {(x^x)^{(x^x)^{(x^x)\\\\ldots}} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le a \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"incomplete gamma 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