Type 2 Problems Part 8 Problem No 8 - Definite Integration - Diploma Maths II

TL;DR

The video demonstrates how to solve a definite integral problem using Property Number 2, resulting in the answer PI square divided by 4.

Transcript

click the bell icon to get latest videos from Ekeeda Hello friends in this video we are going to continue with one more problem which is based on property number 2 of definite integral so let us start with problem number 8 integral 0 to Pie X into sine 4x DX let us use the property first where we are going to write X as upper limit Pie minus X so w... Read More

Key Insights

• ➖ Property Number 2 states that the integral of a function from 0 to a is equal to the integral of the same function with its argument replaced by a minus x.
• ❓ The value of sin(PI - X) remains the same as sin(X) because PI is an even multiple of 90 degrees (180 degrees).
• 👨‍💼 The sine function is positive in the second quadrant.
• 🥳 The integral problem can be simplified by separating the integral into two parts and using the half angle formula for sine squared.

Questions & Answers

Q: What is Property Number 2 of definite integrals?

Property Number 2 states that the integral of f(x) from 0 to a is equal to the integral of f(a-x) from 0 to a.

Q: How does one determine the value of sin(PI - X)?

Since PI is an even multiple of 90 degrees (180 degrees), the value of sin(PI - X) remains the same as sin(X). We do not need to change the value of the sine function.

Q: In which quadrant does the angle PI - X lie?

The angle PI - X lies in the second quadrant. In this quadrant, the sine function is positive.

Q: What is the final answer to the integral problem?

The final answer to the integral problem is PI square divided by 4.

Summary & Key Takeaways

• The video discusses solving a definite integral problem using Property Number 2, which states that 0 to a f(x) dx is equal to integral 0 to a f(a-x) dx.

• The problem given is integral 0 to PI x sin(4x) dx.

• By applying Property Number 2 and simplifying the equation, the final answer is determined to be PI square divided by 4.