Integral of the Day 10.24.23 | Rational Function--but NOT partial fractions| Math with Professor V

Although my class this semester has now moved on to studying sequences and series, I still thought it would be fun to record solving this integral! I feel like they're great daily brain puzzles. I used to hate these tricky rational functions with denominators that didn't factor, but now I really enjoy them. What about you?

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xoxo,
Professor V

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[georgesadler]

Professor V, thank you for a challenging Integral of the Day which involves a Rational Function. Professor V, I did this integral in a different way in comparison to the way you computed yours from start to finish. I factor out a nine in the denominator and then I completed the square and the integral becomes 1/9 times the integral of (x +1) dx/ (x +1/3)^2 + 4/9. Professor V, I let u equal to x +1/3, which breaks up the integral into two pieces. The two integrals are 1/9 times the integral of u du/ u squared plus 4/9 and 2/27 times the integral of du/(u squared + 4/9). Professor V, the solution to these two integrals are natural logs and Arctangent. Professor V, my final solution to this integral is 1/18 ln (x squared +(2/3 )x + 5/9) + 1/9 Arctan ((3x +1)/2) plus a Constant. Professor V, I differentiated the solution, and I got back the function that I started with in the beginning of this video. This is an error free video/lecture on YouTube TV with Math TV Professor V.