{"id":45916,"date":"2025-07-01T14:26:31","date_gmt":"2025-07-01T14:26:31","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=45916"},"modified":"2025-07-01T14:26:33","modified_gmt":"2025-07-01T14:26:33","slug":"integrate-2","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/integrate-2\/","title":{"rendered":"Integrate"},"content":{"rendered":"\n

Integrate: \u00e2\u02c6\u00ab csc(x) cot(x) – sec^2(x) dx<\/p>\n\n\n\n

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

To solve the integral \u222bcsc\u2061(x)cot\u2061(x)\u2212sec\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) – \\sec^2(x) \\, dx\u222bcsc(x)cot(x)\u2212sec2(x)dx, we will break it into two simpler integrals.<\/p>\n\n\n\n

Step 1: Separate the integral<\/h3>\n\n\n\n

We can split the integral as follows:\u222bcsc\u2061(x)cot\u2061(x)\u2212sec\u20612(x)\u2009dx=\u222bcsc\u2061(x)cot\u2061(x)\u2009dx\u2212\u222bsec\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) – \\sec^2(x) \\, dx = \\int \\csc(x) \\cot(x) \\, dx – \\int \\sec^2(x) \\, dx\u222bcsc(x)cot(x)\u2212sec2(x)dx=\u222bcsc(x)cot(x)dx\u2212\u222bsec2(x)dx<\/p>\n\n\n\n

Step 2: Solve the first integral \u222bcsc\u2061(x)cot\u2061(x)\u2009dx\\int \\csc(x) \\cot(x) \\, dx\u222bcsc(x)cot(x)dx<\/h3>\n\n\n\n

Recall that the derivative of csc\u2061(x)\\csc(x)csc(x) is \u2212csc\u2061(x)cot\u2061(x)-\\csc(x) \\cot(x)\u2212csc(x)cot(x), which is useful for recognizing the integral.<\/p>\n\n\n\n

The integral of csc\u2061(x)cot\u2061(x)\\csc(x) \\cot(x)csc(x)cot(x) is simply \u2212csc\u2061(x)-\\csc(x)\u2212csc(x), since:\u222bcsc\u2061(x)cot\u2061(x)\u2009dx=\u2212csc\u2061(x)\\int \\csc(x) \\cot(x) \\, dx = -\\csc(x)\u222bcsc(x)cot(x)dx=\u2212csc(x)<\/p>\n\n\n\n

Step 3: Solve the second integral \u222bsec\u20612(x)\u2009dx\\int \\sec^2(x) \\, dx\u222bsec2(x)dx<\/h3>\n\n\n\n

The integral of sec\u20612(x)\\sec^2(x)sec2(x) is a standard result:\u222bsec\u20612(x)\u2009dx=tan\u2061(x)\\int \\sec^2(x) \\, dx = \\tan(x)\u222bsec2(x)dx=tan(x)<\/p>\n\n\n\n

Step 4: Combine the results<\/h3>\n\n\n\n

Now, we can combine the results from both integrals:\u222bcsc\u2061(x)cot\u2061(x)\u2212sec\u20612(x)\u2009dx=\u2212csc\u2061(x)\u2212tan\u2061(x)+C\\int \\csc(x) \\cot(x) – \\sec^2(x) \\, dx = -\\csc(x) – \\tan(x) + C\u222bcsc(x)cot(x)\u2212sec2(x)dx=\u2212csc(x)\u2212tan(x)+C<\/p>\n\n\n\n

Where CCC is the constant of integration.<\/p>\n\n\n\n

Final Answer:<\/h3>\n\n\n\n

\u2212csc\u2061(x)\u2212tan\u2061(x)+C-\\csc(x) – \\tan(x) + C\u2212csc(x)\u2212tan(x)+C<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n
    \n
  • The first integral \u222bcsc\u2061(x)cot\u2061(x)\u2009dx\\int \\csc(x) \\cot(x) \\, dx\u222bcsc(x)cot(x)dx was recognized by its similarity to the derivative of csc\u2061(x)\\csc(x)csc(x).<\/li>\n\n\n\n
  • The second integral \u222bsec\u20612(x)\u2009dx\\int \\sec^2(x) \\, dx\u222bsec2(x)dx was solved using the standard formula for the derivative of tan\u2061(x)\\tan(x)tan(x).<\/li>\n\n\n\n
  • By combining both, the result was obtained.<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Integrate: \u00e2\u02c6\u00ab csc(x) cot(x) – sec^2(x) dx The Correct Answer and Explanation is: To solve the integral \u222bcsc\u2061(x)cot\u2061(x)\u2212sec\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) – \\sec^2(x) \\, dx\u222bcsc(x)cot(x)\u2212sec2(x)dx, we will break it into two simpler integrals. Step 1: Separate the integral We can split the integral as follows:\u222bcsc\u2061(x)cot\u2061(x)\u2212sec\u20612(x)\u2009dx=\u222bcsc\u2061(x)cot\u2061(x)\u2009dx\u2212\u222bsec\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) – \\sec^2(x) \\, dx = \\int \\csc(x) \\cot(x) \\, […]<\/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-45916","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45916","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=45916"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45916\/revisions"}],"predecessor-version":[{"id":45918,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45916\/revisions\/45918"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=45916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=45916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=45916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}