fun integral battle#1: thank you trig identities  Summary and Q&A
TL;DR
Two integrals are shown, with the first one being easier to solve due to recognizable terms and cancellation of factors.
Questions & Answers
Q: Why is the first integral considered easier to solve?
The first integral is easier because it involves recognizable terms and a cancellation of factors. Recognizing that 1 plus cosine squared x is the same as (cosine x)^2 allows for the cancellation of a sine x term, making the integral simpler to solve.
Q: What substitution is used to simplify the first integral?
The substitution used is letting u equal to cosine x. This transforms the integral into the "U world" where the cancellation of the sine x term becomes apparent.
Q: How is the second integral simplified?
The second integral is simplified by utilizing the double angle identity for sine. By rewriting sine of 2x as 2 times sine x times cosine x, the integral becomes expressible in terms of x inside sine or cosine functions.
Q: What substitution is used to simplify the second integral?
In this case, the substitution used is letting u equal to the entire denominator, 1 plus cosine squared x. Transforming the integral into the "U world" allows for cancellation of terms, ultimately resulting in a negative natural logarithm function.
Summary & Key Takeaways

The content presents two integrals: the integral of sine of 2x over 1 plus cosine squared x, and the integral of only x over 1 plus cosine squared x.

The first integral is considered easier due to the recognition of terms and cancellation of factors.

The integrals are simplified through substitution, with the first integral being transformed into the "U world" and the second integral utilizing the double angle identity for sine.