What Is the Integral of sin(nx) from 0 to pi?

TL;DR

The integral of sin(nx) from 0 to pi is 0 for even values of n and equals 2/n for odd values of n. This pattern shows that the areas under the sine curve cancel out for even n while providing a specific value for odd n, confirming via differentiation that the relationship holds.

Transcript

in this problem we have to integrate the sine of nx with respect to x from 0 to pi and we have to see if we can find a pattern for values of n equals 0 1 2 3 and so on so when you integrate the sine of nx you end up getting the negative cosine of nx divided by n so because we have to divide by n let's take a look at the case when n equals 0 first s... Read More

Key Insights

• ☺️ The integral of the sine function with respect to x from 0 to pi follows a pattern depending on the value of n.
• 👨‍💼 When n is even, the integral is 0, indicating that the positive and negative areas of the sine curve cancel each other out over a complete cycle.
• 🦕 When n is odd, the integral simplifies to 2 divided by n, suggesting that the area under the curve increases proportionally with n.
• ❓ Differentiating the integrated result confirms the validity of the pattern.

Summary & Key Takeaways

• The video demonstrates the process of integrating the sine function with respect to x from 0 to pi.

• The pattern for the integral is found to be negative cosine of nx divided by n, where n represents different values (0, 1, 2, 3, etc.).

• When n is 0, the integral is 0. When n is odd, the integral simplifies to 2 divided by n. When n is even, the integral is 0.