{“appState”:{“pageLoadApiCallsStatus”:true},”articleState”:{“article”:{“headers”:{“creationTime”:”2016-03-26T21:40:54+00:00″,”modifiedTime”:”2016-03-26T21:40:54+00:00″,”timestamp”:”2022-09-14T18:10:18+00:00″},”data”:{“breadcrumbs”:[{“name”:”Academics & The Arts”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33662″},”slug”:”academics-the-arts”,”categoryId”:33662},{“name”:”Math”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33720″},”slug”:”math”,”categoryId”:33720},{“name”:”Calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33723″},”slug”:”calculus”,”categoryId”:33723}],”title”:”How to Use Tangent Substitution to Integrate”,”strippedTitle”:”how to use tangent substitution to integrate”,”slug”:”how-to-use-tangent-substitution-to-integrate”,”canonicalUrl”:””,”seo”:{“metaDescription”:”With the trigonometric substitution method, you can do integrals containing radicals of the following forms: where a is a constant and u is an expression contai”,”noIndex”:0,”noFollow”:0},”content”:”<p>With the trigonometric substitution method, you can do integrals containing radicals of the following forms: </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179883.image0.png\” width=\”336\” height=\”42\” alt=\”image0.png\”/>\n<p>where <i>a</i> is a constant and <i>u</i> is an expression containing <i>x</i>. </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179884.image1.png\” width=\”446\” height=\”42\” alt=\”image1.png\”/>\n<p>You’re going to love this technique … about as much as sticking a hot poker in your eye.</p>\n<p>Before you look at how trigonometric substitution works, here are some mnemonic tricks to help you keep this method straight. Remember, with mnemonic devices, silly (and vulgar) works. First, this involves three trig functions, tangent, sine, and secant. Their initial letters, t, s, and s, are the same letters as the initial letters of the name of this technique, <u>t</u>rigonometric <u>s</u>ubstitution. Pretty nice, eh?</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179885.image2.png\” width=\”490\” height=\”274\” alt=\”image2.png\”/>\n<p>Okay, this is admittedly pretty weak. If you can come up with a better mnemonic, use it!</p>\n<p>Now, ready to do some problems? </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179886.image3.png\” width=\”428\” height=\”196\” alt=\”image3.png\”/>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>1. Draw a right triangle—basically a <i>SohCahToa</i> triangle—where </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179887.image4.png\” width=\”484\” height=\”442\” alt=\”image4.png\”/>\n<img src=\”https://www.dummies.com/wp-content/uploads/179888.image5.jpg\” width=\”523\” height=\”354\” alt=\”image5.jpg\”/>\n </li>\n <li><p class=\”first-para\”>Solve </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179889.image6.png\” width=\”174\” height=\”66\” alt=\”image6.png\”/>\n<p class=\”child-para\”>differentiate, and solve for <i>dx</i>.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179890.image7.png\” width=\”178\” height=\”298\” alt=\”image7.png\”/>\n </li>\n <li><p class=\”first-para\”>Find which trig function is represented by the radical over the <i>a</i>, and then solve for the radical.</p>\n<p class=\”child-para\”>Look at the triangle in the figure. The radical is the hypotenuse and <i>a</i> is 2, the <i>adjacent</i> side, so </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179891.image8.png\” width=\”376\” height=\”188\” alt=\”image8.png\”/>\n </li>\n <li><p class=\”first-para\”>Use the results from Steps 2 and 3 to make substitutions in the original problem and then integrate.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179892.image9.png\” width=\”484\” height=\”376\” alt=\”image9.png\”/>\n </li>\n <li><p class=\”first-para\”>Substitute the <i>x</i> expressions from Steps 1 and 3 back in for </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179893.image10.png\” width=\”174\” height=\”32\” alt=\”image10.png\”/>\n<p class=\”child-para\”>You can also get the expressions from the triangle in the above figure.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179894.image11.png\” width=\”402\” height=\”476\” alt=\”image11.png\”/>\n </li>\n</ol>”,”description”:”<p>With the trigonometric substitution method, you can do integrals containing radicals of the following forms: </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179883.image0.png\” width=\”336\” height=\”42\” alt=\”image0.png\”/>\n<p>where <i>a</i> is a constant and <i>u</i> is an expression containing <i>x</i>. </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179884.image1.png\” width=\”446\” height=\”42\” alt=\”image1.png\”/>\n<p>You’re going to love this technique … about as much as sticking a hot poker in your eye.</p>\n<p>Before you look at how trigonometric substitution works, here are some mnemonic tricks to help you keep this method straight. Remember, with mnemonic devices, silly (and vulgar) works. First, this involves three trig functions, tangent, sine, and secant. Their initial letters, t, s, and s, are the same letters as the initial letters of the name of this technique, <u>t</u>rigonometric <u>s</u>ubstitution. Pretty nice, eh?</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179885.image2.png\” width=\”490\” height=\”274\” alt=\”image2.png\”/>\n<p>Okay, this is admittedly pretty weak. If you can come up with a better mnemonic, use it!</p>\n<p>Now, ready to do some problems? </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179886.image3.png\” width=\”428\” height=\”196\” alt=\”image3.png\”/>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>1. Draw a right triangle—basically a <i>SohCahToa</i> triangle—where </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179887.image4.png\” width=\”484\” height=\”442\” alt=\”image4.png\”/>\n<img src=\”https://www.dummies.com/wp-content/uploads/179888.image5.jpg\” width=\”523\” height=\”354\” alt=\”image5.jpg\”/>\n </li>\n <li><p class=\”first-para\”>Solve </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179889.image6.png\” width=\”174\” height=\”66\” alt=\”image6.png\”/>\n<p class=\”child-para\”>differentiate, and solve for <i>dx</i>.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179890.image7.png\” width=\”178\” height=\”298\” alt=\”image7.png\”/>\n </li>\n <li><p class=\”first-para\”>Find which trig function is represented by the radical over the <i>a</i>, and then solve for the radical.</p>\n<p class=\”child-para\”>Look at the triangle in the figure. The radical is the hypotenuse and <i>a</i> is 2, the <i>adjacent</i> side, so </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179891.image8.png\” width=\”376\” height=\”188\” alt=\”image8.png\”/>\n </li>\n <li><p class=\”first-para\”>Use the results from Steps 2 and 3 to make substitutions in the original problem and then integrate.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179892.image9.png\” width=\”484\” height=\”376\” alt=\”image9.png\”/>\n </li>\n <li><p class=\”first-para\”>Substitute the <i>x</i> expressions from Steps 1 and 3 back in for </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179893.image10.png\” width=\”174\” height=\”32\” alt=\”image10.png\”/>\n<p class=\”child-para\”>You can also get the expressions from the triangle in the above figure.</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/179894.image11.png\” width=\”402\” height=\”476\” alt=\”image11.png\”/>\n </li>\n</ol>”,”blurb”:””,”authors”:[],”primaryCategoryTaxonomy”:{“categoryId”:33723,”title”:”Calculus”,”slug”:”calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33723″}},”secondaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”tertiaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”trendingArticles”:null,”inThisArticle”:[],”relatedArticles”:{“fromBook”:[],”fromCategory”:[{“articleId”:256336,”title”:”Solve a Difficult Limit Problem Using the Sandwich Method”,”slug”:”solve-a-difficult-limit-problem-using-the-sandwich-method”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/256336″}},{“articleId”:255765,”title”:”Solve Limit Problems on a Calculator Using Graphing 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