{"id":703,"date":"2025-06-20T17:06:14","date_gmt":"2025-06-20T17:06:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=703"},"modified":"2025-09-10T14:15:50","modified_gmt":"2025-09-10T14:15:50","slug":"trigonometric-integrals-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-learn-it-3\/","title":{"raw":"Trigonometric Integrals: Learn It 3","rendered":"Trigonometric Integrals: Learn It 3"},"content":{"raw":"

Reduction Formulas<\/h2>\r\nEvaluating [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] for values of [latex]n[\/latex] where [latex]n[\/latex] is odd requires integration by parts. In addition, we must also know the value of [latex]{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex] to evaluate [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex]. The evaluation of [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex] also requires being able to integrate [latex]{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]. To make the process easier, we can derive and apply the following power reduction formulas<\/span>. These rules allow us to replace the integral of a power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex] with the integral of a lower power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex].\r\n\r\n
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reduction formulas for [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] and [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex]<\/h3>\r\n
[latex]{\\displaystyle\\int}{\\sec}^{n}xdx=\\frac{1}{n - 1}{\\sec}^{n - 2}x\\tan{x}+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex]<\/center> \r\n\r\n
[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/center><\/div>\r\n<\/section>
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Apply a reduction formula to evaluate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44553899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44553899\"]\r\n

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By applying the first reduction formula, we obtain<\/p>\r\n\r\n

[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\sec}^{3}xdx\\hfill & =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}{\\displaystyle\\int}\\sec{x}dx\\hfill \\\\ \\hfill & =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}\\text{ln}|\\sec{x}+\\tan{x}|+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>
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Evaluate [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44552899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44552899\"]\r\n

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Applying the reduction formula for [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex] we have<\/p>\r\n\r\n

[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\tan}^{4}xdx& =\\frac{1}{3}{\\tan}^{3}x-{\\displaystyle\\int}{\\tan}^{2}xdx\\hfill & & \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\left(\\tan{x}-{\\displaystyle\\int}{\\tan}^{0}xdx\\right)\\hfill & & \\text{Apply the reduction formula to}{\\displaystyle\\int}{\\tan}^{2}xdx.\\hfill \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+{\\displaystyle\\int}1dx\\hfill & & \\text{Simplify}.\\hfill \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+x+C.\\hfill & & \\text{Evaluate}{\\displaystyle\\int}1dx.\\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>
[ohm_question hide_question_numbers=1]311295[\/ohm_question]<\/section>","rendered":"

Reduction Formulas<\/h2>\n

Evaluating [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] for values of [latex]n[\/latex] where [latex]n[\/latex] is odd requires integration by parts. In addition, we must also know the value of [latex]{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex] to evaluate [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex]. The evaluation of [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex] also requires being able to integrate [latex]{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]. To make the process easier, we can derive and apply the following power reduction formulas<\/span>. These rules allow us to replace the integral of a power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex] with the integral of a lower power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex].<\/p>\n

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reduction formulas for [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] and [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex]<\/h3>\n
[latex]{\\displaystyle\\int}{\\sec}^{n}xdx=\\frac{1}{n - 1}{\\sec}^{n - 2}x\\tan{x}+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex]<\/div>\n

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[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/div>\n<\/div>\n<\/section>\n
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Apply a reduction formula to evaluate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\n<\/div>\n