# Integral of sin(nx) from 0 to pi for n = 0, 1, 2, 3, ... | Summary and Q&A

26.1K views
β’
November 1, 2020
by
The Math Sorcerer
Integral of sin(nx) from 0 to pi for n = 0, 1, 2, 3, ...

## TL;DR

The video explains how to integrate the sine function with respect to x from 0 to pi, finding a pattern for different values of n.

## Key Insights

• βΊοΈ The integral of the sine function with respect to x from 0 to pi follows a pattern depending on the value of n.
• π¨βπΌ 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.
• π¦ When n is odd, the integral simplifies to 2 divided by n, suggesting that the area under the curve increases proportionally with n.
• β Differentiating the integrated result confirms the validity of the pattern.

## 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

### Q: What is the integral of the sine of nx with respect to x from 0 to pi?

The integral of the sine of nx with respect to x from 0 to pi is equal to 0 when n is even and 2 divided by n when n is odd.

### Q: How can the pattern for the integral be derived?

By integrating the sine function, which becomes negative cosine of nx divided by n, and then substituting the limits of integration (0 and pi), the pattern can be observed.

### Q: What happens when n is 0 in the integral?

When n is 0, the definite integral of the sine of nx with respect to x from 0 to pi is equal to 0, as the sine of 0 is 0.

### Q: How can the validity of the pattern be verified?

The pattern can be verified by differentiating the integrated result, which will yield the original sine function with a negative sign due to the division by n.

## 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.
The text passage contains a summary and Q&A about the integral of sin(nx) from 0 to pi for different values of n. It includes multiple questions and their corresponding answers.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Integral of sin(nx) from 0 to pi for n = 0, 1, 2, 3, ... | Summary and Q&A

26.1K views
β’
November 1, 2020
by
The Math Sorcerer
Integral of sin(nx) from 0 to pi for n = 0, 1, 2, 3, ...

## TL;DR

The video explains how to integrate the sine function with respect to x from 0 to pi, finding a pattern for different values of n.

## Key Insights

• βΊοΈ The integral of the sine function with respect to x from 0 to pi follows a pattern depending on the value of n.
• π¨βπΌ 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.
• π¦ When n is odd, the integral simplifies to 2 divided by n, suggesting that the area under the curve increases proportionally with n.
• β Differentiating the integrated result confirms the validity of the pattern.

## 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

### Q: What is the integral of the sine of nx with respect to x from 0 to pi?

The integral of the sine of nx with respect to x from 0 to pi is equal to 0 when n is even and 2 divided by n when n is odd.

### Q: How can the pattern for the integral be derived?

By integrating the sine function, which becomes negative cosine of nx divided by n, and then substituting the limits of integration (0 and pi), the pattern can be observed.

### Q: What happens when n is 0 in the integral?

When n is 0, the definite integral of the sine of nx with respect to x from 0 to pi is equal to 0, as the sine of 0 is 0.

### Q: How can the validity of the pattern be verified?

The pattern can be verified by differentiating the integrated result, which will yield the original sine function with a negative sign due to the division by n.

## 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.
user
The text passage contains a summary and Q&A about the integral of sin(nx) from 0 to pi for different values of n. It includes multiple questions and their corresponding answers.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS