{"id":45,"date":"2024-04-17T11:43:57","date_gmt":"2024-04-17T15:43:57","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?post_type=chapter&p=45"},"modified":"2024-10-31T20:08:33","modified_gmt":"2024-11-01T00:08:33","slug":"trigonometry-substitution-for-indefinite-integrals","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/trigonometry-substitution-for-indefinite-integrals\/","title":{"raw":"Unit 7: Trigonometry Substitution for Indefinite Integrals","rendered":"Unit 7: Trigonometry Substitution for Indefinite Integrals"},"content":{"raw":"Trig substitution is a technique used in calculus to evaluate indefinite integrals that involve expressions with square roots, particularly those of the form [latex]\\sqrt{a^2-b^2x^2}, \\sqrt{a^2+b^2x^2}, [\/latex] [latex]\\sqrt{b^2x^2-a^2}[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. Trigonometric substitution involves replacing these expressions with trigonometric identities \u2014 such as [latex]x=\\frac{a}{b}\\sin\\theta, [\/latex] [latex]x=\\frac{a}{b}\\tan\\theta, x=\\frac{a}{b}\\sec\\theta[\/latex] <\/span><\/span><\/span>\u2014 we can simplify these expressions into forms that are easier to integrate.\r\n\r\nThe process of trigonometric substitution typically involves three main steps.\r\n\r\n(1) Choose the appropriate trig substitution<\/strong>: Identify the form of the square root expression in the integral, and select the corresponding trig substitution (refer to the table below).\r\n\r\n(2) Rewrite the integral in terms of [latex]\\theta[\/latex]: <\/strong><\/strong>replace [latex]x[\/latex] with the chosen trig substitution. Simplify the integral using trig identities to eliminate square roots, which often reduces the integral to a simpler trigonometric form that\u2019s easier to evaluate.<\/span>\r\n\r\n(3) Integrate and Back-Substitute<\/strong>: Evaluate the integral in terms of [latex]\\theta[\/latex]. Once you have the antiderivative, use inverse trigonometric functions to convert [latex]\\theta[\/latex] back to the original variable [latex]x[\/latex], yielding the final solution in terms of\u00a0 [latex]x[\/latex].<\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
square root expression<\/strong><\/td>\r\nsubstitution<\/strong><\/td>\r\nbounds of [latex]\\theta[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
[latex]\\sqrt{a^2-b^2x^2}[\/latex]<\/td>\r\n[latex]x=\\frac{a}{b}\\sin\\theta[\/latex]<\/td>\r\n[latex]\\theta \\in [-\\pi\/2, \\pi\/2][\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\sqrt{a^2+b^2x^2}[\/latex]<\/td>\r\n[latex]x=\\frac{a}{b}\\tan\\theta[\/latex]<\/td>\r\n[latex]\\theta \\in [-\\pi\/2, \\pi\/2][\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\sqrt{b^2x^2-a^2}[\/latex]<\/td>\r\n[latex]x=\\frac{a}{b}\\sec\\theta[\/latex]<\/td>\r\n[latex]\\theta \\in [0, \\pi\/2)\\cup(\\pi\/2,\\pi][\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Practice Questions:<\/strong><\/h6>\r\nEvaluate the following indefinite integrals before referring to the video for comprehensive solutions.\r\n\r\nQ1: [latex]\\int \\frac{1}{\\sqrt{4+x^2}}dx[\/latex]\r\n\r\nQ2: [latex]\\int \\frac{x^2}{\\sqrt{9-x^2}}dx[\/latex]","rendered":"

Trig substitution is a technique used in calculus to evaluate indefinite integrals that involve expressions with square roots, particularly those of the form [latex]\\sqrt{a^2-b^2x^2}, \\sqrt{a^2+b^2x^2},[\/latex] [latex]\\sqrt{b^2x^2-a^2}[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. Trigonometric substitution involves replacing these expressions with trigonometric identities \u2014 such as [latex]x=\\frac{a}{b}\\sin\\theta,[\/latex] [latex]x=\\frac{a}{b}\\tan\\theta, x=\\frac{a}{b}\\sec\\theta[\/latex] <\/span><\/span><\/span>\u2014 we can simplify these expressions into forms that are easier to integrate.<\/p>\n

The process of trigonometric substitution typically involves three main steps.<\/p>\n

(1) Choose the appropriate trig substitution<\/strong>: Identify the form of the square root expression in the integral, and select the corresponding trig substitution (refer to the table below).<\/p>\n

(2) Rewrite the integral in terms of [latex]\\theta[\/latex]: <\/strong><\/strong>replace [latex]x[\/latex] with the chosen trig substitution. Simplify the integral using trig identities to eliminate square roots, which often reduces the integral to a simpler trigonometric form that\u2019s easier to evaluate.<\/span><\/p>\n

(3) Integrate and Back-Substitute<\/strong>: Evaluate the integral in terms of [latex]\\theta[\/latex]. Once you have the antiderivative, use inverse trigonometric functions to convert [latex]\\theta[\/latex] back to the original variable [latex]x[\/latex], yielding the final solution in terms of\u00a0 [latex]x[\/latex].<\/span><\/p>\n\n\n\n\n\n\n
square root expression<\/strong><\/td>\nsubstitution<\/strong><\/td>\nbounds of [latex]\\theta[\/latex]<\/strong><\/td>\n<\/tr>\n
[latex]\\sqrt{a^2-b^2x^2}[\/latex]<\/td>\n[latex]x=\\frac{a}{b}\\sin\\theta[\/latex]<\/td>\n[latex]\\theta \\in [-\\pi\/2, \\pi\/2][\/latex]<\/td>\n<\/tr>\n
[latex]\\sqrt{a^2+b^2x^2}[\/latex]<\/td>\n[latex]x=\\frac{a}{b}\\tan\\theta[\/latex]<\/td>\n[latex]\\theta \\in [-\\pi\/2, \\pi\/2][\/latex]<\/td>\n<\/tr>\n
[latex]\\sqrt{b^2x^2-a^2}[\/latex]<\/td>\n[latex]x=\\frac{a}{b}\\sec\\theta[\/latex]<\/td>\n[latex]\\theta \\in [0, \\pi\/2)\\cup(\\pi\/2,\\pi][\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Practice Questions:<\/strong><\/h6>\n

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.<\/p>\n

Q1: [latex]\\int \\frac{1}{\\sqrt{4+x^2}}dx[\/latex]<\/p>\n

Q2: [latex]\\int \\frac{x^2}{\\sqrt{9-x^2}}dx[\/latex]<\/p>\n","protected":false},"author":391,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-45","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/45","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/users\/391"}],"version-history":[{"count":41,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/45\/revisions"}],"predecessor-version":[{"id":180,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/45\/revisions\/180"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/45\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/media?parent=45"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapter-type?post=45"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/contributor?post=45"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/license?post=45"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}