We are given the improper integral: I=\u222b1\u221e8×8\u2009dxI = \\int_1^{\\infty} \\frac{8}{\\sqrt[8]{x}} \\, dxI=\u222b1\u221e\u200b8x\u200b8\u200bdx<\/p>\n\n\n\n
This is an improper integral because the upper limit of integration is infinity. To determine whether the integral converges or diverges, we must first express the integrand in a simpler form and evaluate the integral.<\/p>\n\n\n\n
Step 1: Simplify the integrand<\/h3>\n\n\n\n
Recall that x8=x1\/8\\sqrt[8]{x} = x^{1\/8}8x\u200b=x1\/8. Therefore, the integrand becomes: 8×8=8x\u22121\/8\\frac{8}{\\sqrt[8]{x}} = 8 x^{-1\/8}8x\u200b8\u200b=8x\u22121\/8<\/p>\n\n\n\n
Thus, the integral becomes: I=\u222b1\u221e8x\u22121\/8\u2009dxI = \\int_1^{\\infty} 8 x^{-1\/8} \\, dxI=\u222b1\u221e\u200b8x\u22121\/8dx<\/p>\n\n\n\n
Step 2: Find the antiderivative<\/h3>\n\n\n\n
Now, we can evaluate the indefinite integral. The general rule for integrating xnx^nxn (where n\u2260\u22121n \\neq -1n\ue020=\u22121) is: \u222bxn\u2009dx=xn+1n+1+C\\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C\u222bxndx=n+1xn+1\u200b+C<\/p>\n\n\n\n
For our integral, the exponent is \u22121\/8-1\/8\u22121\/8. Applying the rule, we get: \u222bx\u22121\/8\u2009dx=x7\/87\/8=87×7\/8\\int x^{-1\/8} \\, dx = \\frac{x^{7\/8}}{7\/8} = \\frac{8}{7} x^{7\/8}\u222bx\u22121\/8dx=7\/8×7\/8\u200b=78\u200bx7\/8<\/p>\n\n\n\n
Thus, the antiderivative of 8x\u22121\/88 x^{-1\/8}8x\u22121\/8 is: 8\u22c587×7\/8=647×7\/88 \\cdot \\frac{8}{7} x^{7\/8} = \\frac{64}{7} x^{7\/8}8\u22c578\u200bx7\/8=764\u200bx7\/8<\/p>\n\n\n\n
Step 3: Evaluate the improper integral<\/h3>\n\n\n\n
Now, we evaluate the improper integral from 1 to infinity. We compute the limit of the integral as the upper limit approaches infinity: I=lim\u2061b\u2192\u221e\u222b1b8x\u22121\/8\u2009dx=lim\u2061b\u2192\u221e[647×7\/8]1bI = \\lim_{b \\to \\infty} \\int_1^b 8 x^{-1\/8} \\, dx = \\lim_{b \\to \\infty} \\left[ \\frac{64}{7} x^{7\/8} \\right]_1^bI=b\u2192\u221elim\u200b\u222b1b\u200b8x\u22121\/8dx=b\u2192\u221elim\u200b[764\u200bx7\/8]1b\u200b<\/p>\n\n\n\n
Substituting the limits of integration: I=lim\u2061b\u2192\u221e(647b7\/8\u2212647\u22c517\/8)I = \\lim_{b \\to \\infty} \\left( \\frac{64}{7} b^{7\/8} – \\frac{64}{7} \\cdot 1^{7\/8} \\right)I=b\u2192\u221elim\u200b(764\u200bb7\/8\u2212764\u200b\u22c517\/8) I=lim\u2061b\u2192\u221e647(b7\/8\u22121)I = \\lim_{b \\to \\infty} \\frac{64}{7} \\left( b^{7\/8} – 1 \\right)I=b\u2192\u221elim\u200b764\u200b(b7\/8\u22121)<\/p>\n\n\n\n
As b\u2192\u221eb \\to \\inftyb\u2192\u221e, b7\/8\u2192\u221eb^{7\/8} \\to \\inftyb7\/8\u2192\u221e. Therefore, the expression diverges to infinity.<\/p>\n\n\n\n
Conclusion<\/h3>\n\n\n\n
Since the limit diverges, the improper integral diverges<\/strong>. Thus, the correct answer is: DIVERGES\\boxed{DIVERGES}DIVERGES\u200b<\/p>\n\n\n\n Determine whether the improper integral diverges or converges. \\int_1^(\\infty ) (8)\/(\\root(8)(x))dx Evaluate the integral if it converges. (If the quantity diverges, enter DIVERGES.) The Correct Answer and Explanation is: We are given the improper integral: I=\u222b1\u221e8×8\u2009dxI = \\int_1^{\\infty} \\frac{8}{\\sqrt[8]{x}} \\, dxI=\u222b1\u221e\u200b8x\u200b8\u200bdx This is an improper integral because the upper limit of integration is infinity. To […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-266015","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266015","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=266015"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266015\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266015"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266015"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-1528.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"