How to Integrate Sine Squared of x Step-by-Step

TL;DR

To integrate sine squared of x, use the identity sin²(x) = (1 - cos(2x))/2 to simplify the integral. The result is (1/2)x - (1/4)sin(2x) + C, where you integrate each term separately. This method highlights the importance of trigonometric identities in integration.

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

Key Insights

• 👨‍💼 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.
• 👨‍💼 The integral of 1/2 dx simplifies to 1/2x, while the integral of cosine 2x dx simplifies to 1/4 sine 2x.
• 👨‍💼 Integrating sine squared x becomes easier when recognizing the derivative of cosine is sine, allowing for a direct integration.
• 🗂️ Dividing the integral by the coefficient of x simplifies the integration process.
• 🪈 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.

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.