Integral of sin^2(x) | Summary and Q&A

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February 12, 2019
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The Math Sorcerer
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Integral of sin^2(x)

TL;DR

Learn how to integrate sine squared of x by using the identity sine squared x = (1 - cosine 2x)/2 and simplifying the integral.

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Key Insights

  • πŸ‘¨β€πŸ’Ό The identity sine squared x = (1 - cosine 2x)/2 is crucial in integrating sine squared x.
  • 🍳 Breaking down the integral helps simplify the integration process.
  • πŸ‘¨β€πŸ’Ό The integral of 1/2 dx simplifies to 1/2x, while the integral of cosine 2x dx simplifies to 1/4 sine 2x.
  • πŸ‘¨β€πŸ’Ό Integrating sine squared x becomes easier when recognizing the derivative of cosine is sine, allowing for a direct integration.
  • πŸ—‚οΈ Dividing the integral by the coefficient of x simplifies the integration process.
  • πŸͺˆ It is important to understand the relationship between trigonometric identities and integration techniques in order to solve similar problems efficiently.
  • ☺️ The video demonstrates how to integrate sine squared x step-by-step, providing a clear explanation of each stage.

Transcript

integrate sine squared of X solution in order to integrate sine squared of X it's useful to know a very important identity so the sine squared of X is equal to one minus cosine two x all divided by two so the first step in this problem is to rewrite sine squared using this identity so we can write it as one minus cosine two x all divided by two DX ... Read More

Questions & Answers

Q: What is the first step in integrating sine squared x?

The first step is to rewrite sine squared x as (1 - cosine 2x)/2 using the identity.

Q: How do you break down the integral?

The integral is broken down into two parts: 1/2 times the integral of dx and -1/2 times the integral of cosine 2x dx.

Q: How do you integrate 1/2 dx?

The integration of 1/2 dx simplifies to 1/2x, as the derivative of x is 1.

Q: How do you integrate cosine 2x dx?

To integrate cosine 2x dx, you can use the identity: the integral of cosine x dx is equal to sine x + C. In this case, it becomes 1/4 sine 2x + C.

Summary & Key Takeaways

  • The first step is to rewrite sine squared x using the identity (1 - cosine 2x)/2.

  • Break down the integral into two parts: 1/2 times the integral of dx, and -1/2 times the integral of cosine 2x dx.

  • The integration of 1/2 dx simplifies to 1/2x, and the integration of cosine 2x dx simplifies to 1/4 sine 2x.

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