{"id":32276,"date":"2025-06-22T12:15:36","date_gmt":"2025-06-22T12:15:36","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=32276"},"modified":"2025-06-22T12:15:39","modified_gmt":"2025-06-22T12:15:39","slug":"evaluate-the-following-integral-using-integration-by-parts","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/evaluate-the-following-integral-using-integration-by-parts\/","title":{"rendered":"Evaluate the following integral using integration by parts."},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"506\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-196.png\" alt=\"\" class=\"wp-image-32277\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-196.png 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-196-300x148.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-196-768x380.png 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/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 integral\u222b8xln\u2061(3x)\u2009dx\\int 8x \\ln(3x) \\, dx\u222b8xln(3x)dx<\/p>\n\n\n\n<p>we will use <strong>integration by parts<\/strong>. The formula for integration by parts is:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv &#8211; \\int v \\, du\u222budv=uv\u2212\u222bvdu<\/p>\n\n\n\n<p>Let us choose:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>u=ln\u2061(3x)u = \\ln(3x)u=ln(3x)<\/li>\n\n\n\n<li>dv=8x\u2009dxdv = 8x \\, dxdv=8xdx<\/li>\n<\/ul>\n\n\n\n<p>Now differentiate and integrate:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>du=1x\u22c53\u2009dx=3x\u2009dxdu = \\frac{1}{x} \\cdot 3 \\, dx = \\frac{3}{x} \\, dxdu=x1\u200b\u22c53dx=x3\u200bdx<\/li>\n\n\n\n<li>v=\u222b8x\u2009dx=4x2v = \\int 8x \\, dx = 4x^2v=\u222b8xdx=4&#215;2<\/li>\n<\/ul>\n\n\n\n<p>Now apply the formula:\u222b8xln\u2061(3x)\u2009dx=4x2ln\u2061(3x)\u2212\u222b4&#215;2\u22c53x\u2009dx\\int 8x \\ln(3x) \\, dx = 4x^2 \\ln(3x) &#8211; \\int 4x^2 \\cdot \\frac{3}{x} \\, dx\u222b8xln(3x)dx=4x2ln(3x)\u2212\u222b4&#215;2\u22c5x3\u200bdx=4x2ln\u2061(3x)\u2212\u222b12x\u2009dx= 4x^2 \\ln(3x) &#8211; \\int 12x \\, dx=4x2ln(3x)\u2212\u222b12xdx=4x2ln\u2061(3x)\u22126&#215;2+C= 4x^2 \\ln(3x) &#8211; 6x^2 + C=4x2ln(3x)\u22126&#215;2+C<\/p>\n\n\n\n<p>So, the final answer is:\u222b8xln\u2061(3x)\u2009dx=4x2ln\u2061(3x)\u22126&#215;2+C\\int 8x \\ln(3x) \\, dx = 4x^2 \\ln(3x) &#8211; 6x^2 + C\u222b8xln(3x)dx=4x2ln(3x)\u22126&#215;2+C<\/p>\n\n\n\n<p>Now, looking at the multiple-choice question, the correct setup based on integration by parts is:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>u=ln\u2061(3x)u = \\ln(3x)u=ln(3x)<\/li>\n\n\n\n<li>dv=8x\u2009dxdv = 8x \\, dxdv=8xdx<br>So,<\/li>\n\n\n\n<li>uv=4x2ln\u2061(3x)uv = 4x^2 \\ln(3x)uv=4x2ln(3x)<\/li>\n\n\n\n<li>\u222bv\u2009du=\u222b12x\u2009dx=6&#215;2\\int v \\, du = \\int 12x \\, dx = 6x^2\u222bvdu=\u222b12xdx=6&#215;2<\/li>\n<\/ul>\n\n\n\n<p>The expression should be:4x2ln\u2061(3x)\u2212\u222b12x\u2009dx4x^2 \\ln(3x) &#8211; \\int 12x \\, dx4x2ln(3x)\u2212\u222b12xdx<\/p>\n\n\n\n<p>This matches <strong>Option B<\/strong>:<br><strong>B. 4x2ln\u2061(3x)\u2212\u222b4&#215;2\u2009dx4x^2 \\ln(3x) &#8211; \\int 4x^2 \\, dx4x2ln(3x)\u2212\u222b4x2dx<\/strong><\/p>\n\n\n\n<p>This answer shows the structure of integration by parts where u=ln\u2061(3x)u = \\ln(3x)u=ln(3x) and dv=8x\u2009dxdv = 8x \\, dxdv=8xdx, and sets up a simpler integral.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>The given integral \u222b8xln\u2061(3x)\u2009dx\\int 8x \\ln(3x) \\, dx\u222b8xln(3x)dx is suitable for the integration by parts method because it is the product of an algebraic function and a logarithmic function. Integration by parts helps us integrate such products. The idea is to choose uuu as the function that simplifies when differentiated, which in this case is ln\u2061(3x)\\ln(3x)ln(3x). The remaining part, 8x\u2009dx8x \\, dx8xdx, becomes dvdvdv, which we integrate easily.<\/p>\n\n\n\n<p>We set:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>u=ln\u2061(3x)u = \\ln(3x)u=ln(3x), so du=3xdxdu = \\frac{3}{x} dxdu=x3\u200bdx<\/li>\n\n\n\n<li>dv=8xdxdv = 8x dxdv=8xdx, so v=4x2v = 4x^2v=4&#215;2<\/li>\n<\/ul>\n\n\n\n<p>Then we use the formula \u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u\\,dv = uv &#8211; \\int v\\,du\u222budv=uv\u2212\u222bvdu. This gives:\u222b8xln\u2061(3x)dx=4x2ln\u2061(3x)\u2212\u222b4&#215;2\u22c53xdx\\int 8x \\ln(3x) dx = 4x^2 \\ln(3x) &#8211; \\int 4x^2 \\cdot \\frac{3}{x} dx\u222b8xln(3x)dx=4x2ln(3x)\u2212\u222b4&#215;2\u22c5x3\u200bdx=4x2ln\u2061(3x)\u2212\u222b12xdx=4x2ln\u2061(3x)\u22126&#215;2+C= 4x^2 \\ln(3x) &#8211; \\int 12x dx = 4x^2 \\ln(3x) &#8211; 6x^2 + C=4x2ln(3x)\u2212\u222b12xdx=4x2ln(3x)\u22126&#215;2+C<\/p>\n\n\n\n<p>This confirms Option <strong>B<\/strong> is correct.<\/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-banner8-357.jpeg\" alt=\"\" class=\"wp-image-32278\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-357.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-357-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-357-768x923.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 integral\u222b8xln\u2061(3x)\u2009dx\\int 8x \\ln(3x) \\, dx\u222b8xln(3x)dx we will use integration by parts. The formula for integration by parts is:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv &#8211; \\int v \\, du\u222budv=uv\u2212\u222bvdu Let us choose: Now differentiate and integrate: Now apply the formula:\u222b8xln\u2061(3x)\u2009dx=4x2ln\u2061(3x)\u2212\u222b4&#215;2\u22c53x\u2009dx\\int 8x \\ln(3x) \\, dx = 4x^2 \\ln(3x) [&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-32276","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/32276","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=32276"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/32276\/revisions"}],"predecessor-version":[{"id":32279,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/32276\/revisions\/32279"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=32276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=32276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=32276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}