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Section 5.2 : Computing Indefinite Integrals

20. Evaluate \( \displaystyle \int{{\frac{1}{{1 + {x^2}}} + \frac{{12}}{{\sqrt {1 - {x^2}} }}\,dx}}\).

Show Solution

There really isn’t too much to do other than to evaluate the integral.

\[\int{{\frac{1}{{1 + {x^2}}} + \frac{{12}}{{\sqrt {1 - {x^2}} }}\,dx}} = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\tan }^{ - 1}}\left( x \right) + 12{{\sin }^{ - 1}}\left( x \right) + c}}\]

Note that because of the similarity of the derivative of inverse sine and inverse cosine an alternate answer is,

\[\int{{\frac{1}{{1 + {x^2}}} + \frac{{12}}{{\sqrt {1 - {x^2}} }}\,dx}} = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\tan }^{ - 1}}\left( x \right) - 12{{\cos }^{ - 1}}\left( x \right) + c}}\]