fun integral battle#1: thank you trig identities | Summary and Q&A

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July 6, 2016
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blackpenredpen
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fun integral battle#1: thank you trig identities

TL;DR

Two integrals are shown, with the first one being easier to solve due to recognizable terms and cancellation of factors.

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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.

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