Calculus II For Dummies Cheat Sheet

What my calculus 2 students did to this integral!
What my calculus 2 students did to this integral!

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He is the author of <i>Logic For Dummies</i> and <i>Basic Math &amp; Pre-Algebra For Dummies</i>.”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9399″}}],”_links”:{“self”:”https://dummies-api.dummies.com/v2/books/”}},”collections”:[],”articleAds”:{“footerAd”:”<div class=\”du-ad-region row\” id=\”article_page_adhesion_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_adhesion_ad\” data-refreshed=\”false\” \r\n data-target = \”[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119986614&quot;]}]\” id=\”du-slot-6435a05f8da02\”></div></div>”,”rightAd”:”<div class=\”du-ad-region row\” id=\”article_page_right_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_right_ad\” data-refreshed=\”false\” \r\n data-target = \”[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119986614&quot;]}]\” id=\”du-slot-6435a05f8e2b9\”></div></div>”},”articleType”:{“articleType”:”Cheat Sheet”,”articleList”:[{“articleId”:188540,”title”:”The Most Important Derivatives and Antiderivatives to Know”,”slug”:”the-most-important-derivatives-and-antiderivatives-to-know”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188540″}},{“articleId”:190797,”title”:”The Riemann Sum Formula For the Definite Integral”,”slug”:”the-riemann-sum-formula-for-the-definite-integral”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/190797″}},{“articleId”:188541,”title”:”Integration by Parts with the DI-agonal Method”,”slug”:”integration-by-parts-with-the-di-agonal-method”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188541″}},{“articleId”:188504,”title”:”The Sum Rule, the Constant Multiple Rule, and the Power Rule for Integration”,”slug”:”the-sum-rule-the-constant-multiple-rule-and-the-power-rule-for-integration”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188504″}},{“articleId”:188543,”title”:”How to Solve Integrals with Variable Substitution”,”slug”:”how-to-solve-integrals-with-variable-substitution”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188543″}},{“articleId”:188542,”title”:”How to Use Integration by Parts”,”slug”:”how-to-use-integration-by-parts”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188542″}},{“articleId”:188544,”title”:”How to Solve Compound Functions Where the Inner Function Is <i>ax</i> + <i>b</i>”,”slug”:”how-to-solve-compound-functions-where-the-inner-function-is-ax-b”,”categoryList”:[],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188544″}},{“articleId”:188529,”title”:”Solve Compound Functions Where the Inner Function Is <i>ax</i>”,”slug”:”solve-compound-functions-where-the-inner-function-is-ax”,”categoryList”:[],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/188529″}}],”content”:[{“title”:”The most important derivatives and antiderivatives to know”,”thumb”:null,”image”:null,”content”:”<p>The table below shows you how to differentiate and integrate 18 of the most common functions. As you can see, integration reverses differentiation, returning the function to its original state, up to a constant <i>C</i>.</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246703.image0.png\” alt=\”image0.png\” width=\”401\” height=\”967\” /></p>\n”},{“title”:”The Riemann Sum Formula for the definite integral”,”thumb”:null,”image”:null,”content”:”<p>The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows<i>:</i><i> </i></p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223426.image0.png\” alt=\”image0.png\” width=\”191\” height=\”55\” /></p>\n<p>Below are the steps for approximating an integral using six rectangles:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223427.image1.png\” alt=\”image1.png\” width=\”312\” height=\”36\” /></p>\n<ol class=\”level-one\”>\n<li>\n<p class=\”first-para\”>Increase the number of rectangles (<i>n</i>) to create a better approximation:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223428.image2.png\” alt=\”image2.png\” width=\”259\” height=\”36\” /></li>\n<li>\n<p class=\”first-para\”>Simplify this formula by factoring out <i>w</i> from each term:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223429.image3.png\” alt=\”image3.png\” width=\”239\” height=\”36\” /></li>\n<li>\n<p class=\”first-para\”>Use the summation symbol to make this formula even more compact:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223430.image4.png\” alt=\”image4.png\” width=\”125\” height=\”52\” /></p>\n<p class=\”child-para\”>The value <i>w</i> is the <i>width</i> of each rectangle:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223431.image5.png\” alt=\”image5.png\” width=\”65\” height=\”41\” /></p>\n<p class=\”child-para\”>Each <i>h </i>value<i> </i>is the <i>height</i> of a different rectangle:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223432.image6.png\” alt=\”image6.png\” width=\”35\” height=\”29\” /></p>\n<p class=\”child-para\”>So here is the Riemann Sum formula for <i>approximating</i> an integral using <i>n</i> rectangles:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223433.image7.png\” alt=\”image7.png\” width=\”168\” height=\”52\” /></li>\n<li>\n<p class=\”first-para\”>For a better approximation, use the limit</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223434.image8.png\” alt=\”image8.png\” width=\”51\” height=\”25\” /></li>\n<li>\n<p class=\”first-para\”>to allow the number of rectangles to approach <i>infinity</i>:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/223435.image9.png\” alt=\”image9.png\” width=\”216\” height=\”52\” /></li>\n</ol>\n”},{“title”:”Integration by parts with the DI-agonal method”,”thumb”:null,”image”:null,”content”:”<p>The DI-agonal method is basically integration by parts with a chart that helps you organize information. This method is especially useful when you need to integrate by parts more than once to solve a problem. Use the following table for integration by parts using the DI-agonal method:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246729.image0.jpg\” alt=\”image0.jpg\” width=\”300\” height=\”240\” /></p>\n”},{“title”:”The Sum Rule, the Constant Multiple Rule, and the Power Rule for integration”,”thumb”:null,”image”:null,”content”:”<p>When you perform integration, there are three important rules that you need to know: the Sum Rule, the Constant Multiple Rule, and the Power Rule.</p>\n<p>The Sum Rule for Integration tells you that it’s okay to integrate long expressions term by term. Here it is formally:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246706.image0.png\” alt=\”image0.png\” width=\”239\” height=\”29\” /></p>\n<p>The Constant Multiple Rule for Integration tells you that it’s okay to move a constant outside of an integral before you integrate. Here it is expressed in symbols:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246707.image1.png\” alt=\”image1.png\” width=\”137\” height=\”29\” /></p>\n<p>The Power Rule for Integration allows you to integrate any real power of <i>x</i> (except -1). Here’s the Power Rule expressed formally:</p>\n<p>where <i>n</i> ≠ -1</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246708.image2.png\” alt=\”image2.png\” width=\”120\” height=\”45\” /></p>\n”},{“title”:”How to solve integrals with variable substitution”,”thumb”:null,”image”:null,”content”:”<p>In Calculus, you can use variable substitution to evaluate a complex integral. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work.</p>\n<ol class=\”level-one\”>\n<li>\n<p class=\”first-para\”>Declare a variable <i>u</i>,<i> </i>set it equal to an algebraic expression that appears in the integral, and then substitute <i>u</i> for this expression in the integral.</p>\n</li>\n<li>\n<p class=\”first-para\”>Differentiate <i>u</i> to find</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246732.image0.png\” alt=\”image0.png\” width=\”24\” height=\”41\” /></p>\n<p class=\”child-para\”>and then isolate all <i>x</i> variables on one side of the equal sign.</p>\n</li>\n<li>\n<p class=\”first-para\”>Make another substitution to change <i>dx</i> and all other occurrences of <i>x</i> in the integral to an expression that includes <i>du</i>.</p>\n</li>\n<li>\n<p class=\”first-para\”>Integrate by using <i>u</i> as your new variable of integration.</p>\n</li>\n<li>\n<p class=\”first-para\”>Express this answer in terms of <i>x</i>.</p>\n</li>\n</ol>\n”},{“title”:”How to Use integration by parts”,”thumb”:null,”image”:null,”content”:”<p>When doing Calculus, the formula for integration by parts gives you the option to break down the product of two functions to its factors and integrate it in an altered form. To use integration by parts in Calculus, follow these steps:</p>\n<ol class=\”level-one\”>\n<li>\n<p class=\”first-para\”>Decompose the entire integral (including <i>dx</i>) into two factors.</p>\n</li>\n<li>\n<p class=\”first-para\”>Let the factor without <i>dx </i>equal <i>u</i> and the factor with <i>dx </i>equal <i>dv</i>.</p>\n</li>\n<li>\n<p class=\”first-para\”>Differentiate <i>u</i> to find <i>du</i>, and integrate <i>dv</i> to find <i>v</i>.</p>\n</li>\n<li>\n<p class=\”first-para\”>Use the formula:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246735.image0.png\” alt=\”image0.png\” width=\”123\” height=\”29\” /></li>\n<li>\n<p class=\”first-para\”>Evaluate the right side of this equation to solve the integral.</p>\n</li>\n</ol>\n”},{“title”:”How to solve compound functions where the inner function is ax + b”,”thumb”:null,”image”:null,”content”:”<p>Some integrals of compound functions <i>f</i><i> </i>(<i>g</i>(<i>x</i>)) are easy to do quickly in Calculus. These include compound functions for which you know how to integrate the outer function <i>f</i>, and the inner function <i>g</i>(<i>x</i>) is of the form <i>ax</i> + <i>b</i> — that is, it differentiates to a constant.</p>\n<p>Here are some examples:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246711.image0.png\” alt=\”image0.png\” width=\”231\” height=\”151\” /></p>\n”},{“title”:”Solve compound functions where the inner function is ax”,”thumb”:null,”image”:null,”content”:”<p>When figuring Calculus problems, some integrals of compound functions <i>f</i><i> </i>(<i>g</i>(<i>x</i>)) are easy to do quickly. These include compound functions for which you know how to integrate the outer function <i>f</i>, and the inner function <i>g</i>(<i>x</i>) is of the form <i>ax</i> — that is, it differentiates to a constant.</p>\n<p>Here are some examples:</p>\n<p><img loading=\”lazy\” src=\”https://www.dummies.com/wp-content/uploads/246714.image0.png\” alt=\”image0.png\” width=\”175\” height=\”151\” /></p>\n”}],”videoInfo”:{“videoId”:null,”name”:null,”accountId”:null,”playerId”:null,”thumbnailUrl”:null,”description”:null,”uploadDate”:null}},”sponsorship”:{“sponsorshipPage”:false,”backgroundImage”:{“src”:null,”width”:0,”height”:0},”brandingLine”:””,”brandingLink”:””,”brandingLogo”:{“src”:null,”width”:0,”height”:0},”sponsorAd”:””,”sponsorEbookTitle”:””,”sponsorEbookLink”:””,”sponsorEbookImage”:{“src”:null,”width”:0,”height”:0}},”primaryLearningPath”:”Advance”,”lifeExpectancy”:”Five 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