Verify the integration formulas. ∫(sin^-1 x)^2 d x=x(sin^-1 x)^2-2 x+2 √(1-x^2) sin^-1 x+C

Name: Problem 101, Chapter 7: Thomas Calculus in SI Units
Uploaded: 2023-08-15T22:26:49-08:00
Duration: 4 min 29 s
Channel: Ma. Theresa Alin
Description: Verify the integration formulas. ∫(sin^-1 x)^2 d x=x(sin^-1 x)^2-2 x+2 √(1-x^2) sin^-1 x+C

step 1 For this, probably what we want to verify is the antiderivative of the square of sine inverse of

Verify the integration formulas. ∫(sin^-1 x)^2 d x=x(sin^-1...

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Key Concepts

Integration by Parts

A technique used to integrate products of functions by transforming the original integral into simpler integrals. It is based on the product rule for differentiation and is particularly useful when one function becomes simpler upon differentiation while the other remains manageable upon integration.

Differentiation as a Verification Tool

A method to confirm the correctness of an antiderivative by differentiating it to recover the original integrand. This process relies on the Fundamental Theorem of Calculus and ensures that the integration formula is valid.

Inverse Trigonometric Functions

Functions that 'undo' the standard trigonometric functions, such as arcsin, arccos, and arctan. These functions often exhibit unique behaviors during integration and differentiation, requiring specific techniques to handle their algebraic and calculus properties.

Fundamental Theorem of Calculus

A central principle in calculus that links the operations of differentiation and integration. It states that if a function is continuous over an interval, the integral of its derivative over that interval recovers the original function, thereby providing a means to verify integration formulas.

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