{"id":726,"date":"2025-06-20T17:07:53","date_gmt":"2025-06-20T17:07:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=726"},"modified":"2025-07-17T15:52:49","modified_gmt":"2025-07-17T15:52:49","slug":"other-strategies-for-integration-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/other-strategies-for-integration-learn-it-2\/","title":{"raw":"Other Strategies for Integration: Learn It 2","rendered":"Other Strategies for Integration: Learn It 2"},"content":{"raw":"

Computer Algebra Systems<\/h2>\r\nFor an even faster solution, you might turn to technology. A Computer Algebra System (CAS)<\/strong> is software that can perform symbolic mathematical calculations, including integration, automatically.\r\n

If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.<\/p>\r\n

Computer algebra systems like Wolfram Alpha, Mathematica, or the integration features in graphing calculators can handle complex integrals instantly. However, these tools work best when you understand the underlying techniques\u2014they can solve the problem, but they can't always explain the mathematical reasoning behind each step.<\/p>\r\n\r\n

Use CAS tools to check your work or tackle particularly challenging problems, but don't rely on them exclusively. Understanding the manual techniques helps you recognize when a CAS result makes sense and builds your mathematical intuition.\r\n\r\n<\/section>
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Use a computer algebra system to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Compare this result with [latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C[\/latex], a result we might have obtained if we had used trigonometric substitution.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n

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Using Wolfram Alpha, we obtain<\/p>\r\n\r\n

[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}=\\text{ln}|\\sqrt{{x}^{2}-4}+x|+C[\/latex].<\/div>\r\n \r\n

Notice that<\/p>\r\n\r\n

[latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C=\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}+x}{2}|+C=\\text{ln}|\\sqrt{{x}^{2}-4}+x|-\\text{ln}2+C[\/latex].<\/div>\r\n \r\n

Since these two antiderivatives differ by only a constant, the solutions are equivalent. We could have also demonstrated that each of these antiderivatives is correct by differentiating them.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

Computer Algebra Systems<\/h2>\n

For an even faster solution, you might turn to technology. A Computer Algebra System (CAS)<\/strong> is software that can perform symbolic mathematical calculations, including integration, automatically.<\/p>\n

If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.<\/p>\n

Computer algebra systems like Wolfram Alpha, Mathematica, or the integration features in graphing calculators can handle complex integrals instantly. However, these tools work best when you understand the underlying techniques\u2014they can solve the problem, but they can’t always explain the mathematical reasoning behind each step.<\/p>\n

Use CAS tools to check your work or tackle particularly challenging problems, but don’t rely on them exclusively. Understanding the manual techniques helps you recognize when a CAS result makes sense and builds your mathematical intuition.<\/p>\n<\/section>\n
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Use a computer algebra system to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Compare this result with [latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C[\/latex], a result we might have obtained if we had used trigonometric substitution.<\/p>\n<\/div>\n