{"id":35196,"date":"2025-06-24T08:05:33","date_gmt":"2025-06-24T08:05:33","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=35196"},"modified":"2025-06-24T08:05:35","modified_gmt":"2025-06-24T08:05:35","slug":"find-the-integral-with-respect-to-x-and-simplify","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-integral-with-respect-to-x-and-simplify\/","title":{"rendered":"Find the integral with respect to x and simplify"},"content":{"rendered":"\n

Find the integral with respect to x and simplify: a) \\int \\sin2x\\cos4x\\sin4xdx [3] b) \\int \\frac{2}{x^2\\sqrt{1+x^2}}dx [5] c) \\int \\frac{4x+2}{\\sqrt{x^2-16}}dx [5] d) \\int x^3e^{2x} dx [5] e) \\int \\sqrt{x}\\ln2x dx [5] f) \\int \\sin^{-1}2x dx [5] g) \\int \\cos^43x dx [5] h) \\int \\frac{3x^2+16x+15}{(x+3)^3}dx [5] i) \\int \\frac{3+6x+4x^2-2x^3}{(x^4+3x^2)}dx [5] j) \\int \\frac{x^4-7x-2}{x^3-8}dx [7]<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Certainly, Jeniffer. Let\u2019s tackle these integrals carefully, using the appropriate methods for each. Due to their complexity and your request for a 300-word explanation, I\u2019ll summarize the techniques used, then provide the solutions.<\/p>\n\n\n\n

Overview of Techniques<\/h3>\n\n\n\n

The integrals involve a mix of strategies: trigonometric identities<\/strong>, u-substitution<\/strong>, integration by parts<\/strong>, partial fraction decomposition<\/strong>, and trigonometric powers reduction<\/strong>. Let’s break down the more intricate ones and outline key ideas, focusing on variety and precision over brevity.<\/p>\n\n\n\n

Selected Solutions and Explanation (abridged):<\/h3>\n\n\n\n

a)<\/strong> \u222bsin\u20612xcos\u20614xsin\u20614x\u2009dx\\int \\sin 2x \\cos 4x \\sin 4x \\, dx<\/p>\n\n\n\n

Let\u2019s simplify cos\u20614xsin\u20614x=12sin\u20618x\\cos 4x \\sin 4x = \\frac{1}{2} \\sin 8x. Then:<\/p>\n\n\n\n

\u222bsin\u20612x\u22c512sin\u20618x\u2009dx=12\u222bsin\u20612xsin\u20618x\u2009dx\\int \\sin 2x \\cdot \\frac{1}{2} \\sin 8x \\, dx = \\frac{1}{2} \\int \\sin 2x \\sin 8x \\, dx<\/p>\n\n\n\n

Use product-to-sum:<\/p>\n\n\n\n

sin\u2061Asin\u2061B=12[cos\u2061(A\u2212B)\u2212cos\u2061(A+B)]\\sin A \\sin B = \\frac{1}{2}[\\cos(A-B) – \\cos(A+B)]<\/p>\n\n\n\n

\u21d214[\u222bcos\u2061(6x)\u2212cos\u2061(10x)\u2009dx]=14(sin\u20616×6\u2212sin\u206110×10)+C\\Rightarrow \\frac{1}{4} \\left[ \\int \\cos(6x) – \\cos(10x) \\, dx \\right] = \\frac{1}{4} \\left( \\frac{\\sin 6x}{6} – \\frac{\\sin 10x}{10} \\right) + C<\/p>\n\n\n\n

b)<\/strong> \u222b2×21+x2\u2009dx\\int \\frac{2}{x^2 \\sqrt{1+x^2}} \\, dx<\/p>\n\n\n\n

Let x=tan\u2061\u03b8\u21d2dx=sec\u20612\u03b8d\u03b8x = \\tan \\theta \\Rightarrow dx = \\sec^2 \\theta d\\theta, then it becomes:<\/p>\n\n\n\n

\u222b2tan\u20612\u03b8\u22c5sec\u2061\u03b8\u22c5sec\u20612\u03b8d\u03b8\\int \\frac{2}{\\tan^2 \\theta \\cdot \\sec \\theta} \\cdot \\sec^2 \\theta d\\theta<\/p>\n\n\n\n

Simplify and back-substitute to get:<\/p>\n\n\n\n

\u22122×1+x2+C\\boxed{-\\frac{2}{x \\sqrt{1+x^2}} + C}<\/p>\n\n\n\n

d)<\/strong> \u222bx3e2x\u2009dx\\int x^3 e^{2x} \\, dx<\/p>\n\n\n\n

Use integration by parts repeatedly or tabular method. Final result:<\/p>\n\n\n\n

e2x(x32\u22123×24+3×4\u221238)+C\\boxed{e^{2x} \\left( \\frac{x^3}{2} – \\frac{3x^2}{4} + \\frac{3x}{4} – \\frac{3}{8} \\right) + C}<\/p>\n\n\n\n

The key to mastering integrals lies in recognizing structure and applying substitutions or expansions strategically. If you want full solutions for the remaining ones, I can walk through them too. Shall we continue?<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Find the integral with respect to x and simplify: a) \\int \\sin2x\\cos4x\\sin4xdx [3] b) \\int \\frac{2}{x^2\\sqrt{1+x^2}}dx [5] c) \\int \\frac{4x+2}{\\sqrt{x^2-16}}dx [5] d) \\int x^3e^{2x} dx [5] e) \\int \\sqrt{x}\\ln2x dx [5] f) \\int \\sin^{-1}2x dx [5] g) \\int \\cos^43x dx [5] h) \\int \\frac{3x^2+16x+15}{(x+3)^3}dx [5] i) \\int \\frac{3+6x+4x^2-2x^3}{(x^4+3x^2)}dx [5] j) \\int \\frac{x^4-7x-2}{x^3-8}dx [7] The Correct […]<\/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-35196","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35196","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=35196"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35196\/revisions"}],"predecessor-version":[{"id":35199,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35196\/revisions\/35199"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=35196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=35196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=35196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}