Proofs on Integration of Some Standard Functions Part 4 - Integration - Diploma Maths - II

TL;DR

The video provides a step-by-step proof of the integral DX upon a square minus X square using a logarithmic function.

Transcript

click the bell icon to get latest videos from equator hello friends in this video we are going to see one more proof on integral DX upon a square minus X square let us start in this we need to prove integral DX upon a square minus X square is 1 upon 2 a log a plus X upon a minus X plus C let us consider the given integral as I now the first observa... Read More

Key Insights

• 😑 The proof uses manipulation techniques to simplify and rearrange the integral expression.
• ❎ The formula for a square minus B square is a crucial step in the simplification process.
• 😑 The logarithmic function is employed to integrate the separate terms of the expression.
• 😑 Adjusting the constant factor allows for the correct form of the integral expression.
• 🈸 The proof showcases the application of mathematical properties and formulas to solve integrals.
• 💁 The final result is in the form of log functions and includes a constant of integration.
• ❓ Understanding the properties of logarithmic functions is essential in this proof.

Questions & Answers

Q: What is the purpose of adjusting the constant in the integral expression?

By adjusting the constant, we can obtain the correct form for the integral expression, which includes the 1 upon 2 a factor outside the integral.

Q: How is the denominator simplified using the formula a square minus B square?

The formula a square minus B square is used to simplify the denominator into two separate factors, a minus X and a plus X.

Q: How are the terms in the numerator adjusted to include a positive and negative X?

The terms in the numerator are adjusted by adding X and subtracting X, ensuring both a plus X and a minus X appear in the expression.

Q: How is the integral expression further simplified to obtain the final result?

The integral expression is split into two separate integrals, one for 1 upon a minus X and another for 1 upon a plus X. These integrals can then be evaluated using the logarithmic function.

Summary & Key Takeaways

• The video presents a proof for the integral DX upon a square minus X square.

• The proof involves manipulating the integral expression using formulas and properties of logarithmic functions.

• The final result is 1 upon 2 a log a plus X upon a minus X plus C.