{"id":35654,"date":"2025-06-24T14:48:01","date_gmt":"2025-06-24T14:48:01","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=35654"},"modified":"2025-06-24T14:48:02","modified_gmt":"2025-06-24T14:48:02","slug":"evaluate-the-definite-integral-given-below","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/evaluate-the-definite-integral-given-below\/","title":{"rendered":"Evaluate the definite integral given below."},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"339\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-286.png\" alt=\"\" class=\"wp-image-35655\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-286.png 700w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-286-300x145.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate the definite integral<\/p>\n\n\n\n<p>\u222b24&#215;13(1\u22122x)\u2009dx\\int_{2}^{4} x^{\\frac{1}{3}} (1 &#8211; 2x) \\, dx<\/p>\n\n\n\n<p>we start by expanding the integrand:<\/p>\n\n\n\n<p>x13(1\u22122x)=x13\u22122x43x^{\\frac{1}{3}} (1 &#8211; 2x) = x^{\\frac{1}{3}} &#8211; 2x^{\\frac{4}{3}}<\/p>\n\n\n\n<p>Now integrate term-by-term:<\/p>\n\n\n\n<p>\u222b24(x13\u22122&#215;43)dx=\u222b24x13dx\u22122\u222b24x43dx\\int_{2}^{4} \\left( x^{\\frac{1}{3}} &#8211; 2x^{\\frac{4}{3}} \\right) dx = \\int_{2}^{4} x^{\\frac{1}{3}} dx &#8211; 2 \\int_{2}^{4} x^{\\frac{4}{3}} dx<\/p>\n\n\n\n<p>Use the power rule of integration:<\/p>\n\n\n\n<p>\u222bxn\u2009dx=xn+1n+1+C\\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C<\/p>\n\n\n\n<p>Applying this to each term:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For x13x^{\\frac{1}{3}}, the integral is x4343=34&#215;43\\frac{x^{\\frac{4}{3}}}{\\frac{4}{3}} = \\frac{3}{4} x^{\\frac{4}{3}}<\/li>\n\n\n\n<li>For x43x^{\\frac{4}{3}}, the integral is x7373=37&#215;73\\frac{x^{\\frac{7}{3}}}{\\frac{7}{3}} = \\frac{3}{7} x^{\\frac{7}{3}}<\/li>\n<\/ul>\n\n\n\n<p>Putting it all together:<\/p>\n\n\n\n<p>[34&#215;43]24\u22122[37&#215;73]24=(34\u22c5443\u221234\u22c5243)\u22122(37\u22c5473\u221237\u22c5273)\\left[ \\frac{3}{4} x^{\\frac{4}{3}} \\right]_2^4 &#8211; 2\\left[ \\frac{3}{7} x^{\\frac{7}{3}} \\right]_2^4 = \\left( \\frac{3}{4} \\cdot 4^{\\frac{4}{3}} &#8211; \\frac{3}{4} \\cdot 2^{\\frac{4}{3}} \\right) &#8211; 2\\left( \\frac{3}{7} \\cdot 4^{\\frac{7}{3}} &#8211; \\frac{3}{7} \\cdot 2^{\\frac{7}{3}} \\right)<\/p>\n\n\n\n<p>This evaluates to an exact expression involving rational coefficients and radicals. You can simplify numerically if needed, but unless a decimal is required, this is the exact form.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>This integral tests your ability to combine algebraic simplification with integration techniques. It starts with a product of a power function and a linear expression, prompting distributive expansion. The power rule then allows each term to be integrated independently. Note that x13x^{\\frac{1}{3}} and x43x^{\\frac{4}{3}} are fractional powers, so when applying the rule, increment the exponent and divide by the new exponent. Definite integration simply involves evaluating the antiderivative at the bounds and subtracting.<\/p>\n\n\n\n<p>These types of integrals are essential for modeling non-linear relationships in physics and economics, especially when variables change with fractional powers.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-677.jpeg\" alt=\"\" class=\"wp-image-35656\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-677.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-677-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-677-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The Correct Answer and Explanation is: To evaluate the definite integral \u222b24&#215;13(1\u22122x)\u2009dx\\int_{2}^{4} x^{\\frac{1}{3}} (1 &#8211; 2x) \\, dx we start by expanding the integrand: x13(1\u22122x)=x13\u22122x43x^{\\frac{1}{3}} (1 &#8211; 2x) = x^{\\frac{1}{3}} &#8211; 2x^{\\frac{4}{3}} Now integrate term-by-term: \u222b24(x13\u22122&#215;43)dx=\u222b24x13dx\u22122\u222b24x43dx\\int_{2}^{4} \\left( x^{\\frac{1}{3}} &#8211; 2x^{\\frac{4}{3}} \\right) dx = \\int_{2}^{4} x^{\\frac{1}{3}} dx &#8211; 2 \\int_{2}^{4} x^{\\frac{4}{3}} dx Use the power rule [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-35654","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35654","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=35654"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35654\/revisions"}],"predecessor-version":[{"id":35657,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35654\/revisions\/35657"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=35654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=35654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=35654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}