Proofs on Integration of Some Standard Functions Part 1 - Integration - Diploma Maths - II

TL;DR

This video shows the proof for the integral of DX/√(a^2 - x^2) and establishes its value as sin^(-1)(x/a) + C.

Transcript

click the Bell icon to get latest videos from Ekeeda Hello friends in this video we are going to check proof for integral DX upon under root a square minus X square through that integral DX upon under root a square minus X square is equal to sine inverse X by a plus C in the previous chapter also derivative we have seen certain standard functions f... Read More

Key Insights

• 👍 The integral of DX/√(a^2 - x^2) can be proven using substitution and trigonometric identities.
• ☺️ By substituting X as a sin(theta) and differentiating with respect to theta, the integral is simplified.
• 😑 The simplification leads to the expression of the integral as the sine inverse of (x/a) plus a constant.
• 👨‍💼 The proof demonstrates the connection between the integral of DX/√(a^2 - x^2) and the trigonometric function sine inverse.
• 🎅 The result of sin^(-1)(x/a) + C can be used to solve various mathematical problems involving the given integral.
• 👻 The substitution of X as a sin(theta) allows for the utilization of the identity 1 - sin^2(theta) = cos^2(theta) to simplify the integral.
• ❓ The proof showcases the importance of using substitution and trigonometric identities in solving complex integrals.

Questions & Answers

Q: What is the purpose of the video?

The purpose of the video is to demonstrate the proof for the integral of DX/√(a^2 - x^2) and establish its value as sin^(-1)(x/a) + C.

Q: How is the integral DX/√(a^2 - x^2) simplified using substitution?

The integral is simplified by substituting X as a sin(theta), which allows for the differentiation of X with respect to theta and the evaluation of DX.

Q: What trigonometric identity is used in the proof?

The identity used in the proof is 1 - sin^2(theta) = cos^2(theta), which simplifies the expression for DX/√(a^2 - x^2) to a cos(theta) d(theta)/√(a^2 - cos^2(theta)).

Q: How is the final result of sin^(-1)(x/a) + C derived?

The final result is derived by substituting the value of theta (sin^(-1)(x/a)) into the simplified integral expression and integrating with respect to theta to obtain theta + C, which simplifies to sin^(-1)(x/a) + C.

Summary & Key Takeaways

• The video provides a step-by-step proof for the integral of DX/√(a^2 - x^2) using substitution and trigonometric identities.

• By substituting X as a sin(theta), the video simplifies the integral and differentiates it with respect to theta.

• The final result is found to be sin^(-1)(x/a) + C, proving the integral.