Integral of 1/(1+cos(x) vs integral of sqrt(1+cos(x))

TL;DR

Learn a quick and efficient method for integrating functions involving cosine using simple trigonometric identities.

Transcript

okay between took us on the spot and both of them have the one plus cosine X and in fact how show you guys how to stop this winter gross in just one to two likes so this is going to be a super shortcut to this two particular integrals and I think they do this kind of things a lot on the GE test so the more you know the faster you can perform the ex... Read More

Key Insights

• 😑 Integrating functions with trigonometric expressions can be simplified by recognizing and using trigonometric identities.
• ❓ The integral 1/(1+cosx) can be transformed into 2*cos²(x/2) and further simplified to tangent(x/2).
• ❓ The integral √(1+cosx) can be simplified to 2√2*sin(x/2) by substituting the appropriate trigonometric identity.
• 🐎 Memorizing trigonometric identities can greatly speed up the integration process.
• 🏆 These simplified methods can be particularly useful in exams like the GE test.
• ⏫ Understanding double angle identities for cosine can be applied to simplifying integrals involving cosine.
• 💪 It is important to have a strong knowledge of trigonometric identities to efficiently solve integration problems.

Questions & Answers

Q: How can the integral 1/(1+cosx) be simplified?

By using the trigonometric identity 1+cosx = 2cos²(x/2), the integral can be written as 2cos²(x/2)/(1+cosx). Simplify further to obtain tangent(x/2) as the result.

Q: What is the simplified form of the integral √(1+cosx)?

Recognizing that √(1+cosx) is equivalent to 2√2cos(x/2), the integral simplifies to 2√2sin(x/2).

Q: Are these simplifications applicable to all integrals involving cosine?

No, these simplifications specifically apply to the given integrals 1/(1+cosx) and √(1+cosx). Other integrals involving cosine may require different methods or identities for simplification.

Q: What is the benefit of using these simplified methods for integration?

These methods provide a quick and efficient way to solve integrals involving cosine. By recognizing the appropriate trigonometric identities, the integration process can be significantly simplified.

Summary & Key Takeaways

• The video explains how to simplify the integration of two specific trigonometric functions: 1/(1+cosx) and √(1+cosx).

• For the first integral, 1/(1+cosx), it is shown that it is equivalent to 2*cos²(x/2). By substituting this expression, the integral simplifies to tangent(x/2).

• The second integral, √(1+cosx), is simplified by recognizing it as 2cos(x/2). The integral becomes 2√2sin(x/2).