Integral With Constant Numerator

by Braden
(Botswana)

How do I integrate such integrals with constant numerator and denominator without square root?

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Answer by Pablo:

This integral does not have a square root but falls into the third case of trigonometric substitution. Look it up here: Trigonometric Substitution.

Using the substitution of the third case we make:



Making these substitutions and using the trigonometric identity for the third case:



Now we only need to solve this trigonometric integral and make the inverse substitutions. This trigonometric integral is solved as an example in this page: Trigonometric Integrals. The result is:



Now, substituting back:



And now using the following trigonometric identities:



We have the answer is:



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