Determine the following: ∫0^1 √((x)/(2-x)) d x (Put .x=2 sin^2 θ)

Name: Problem 18, Chapter 30: Engineering Mathematics
Uploaded: 2022-02-17T17:34:36-08:00
Duration: 3 min 44 s
Channel: M Hassan Anwar
Description: Determine the following: ∫0^1 √((x)/(2-x)) d x (Put .x=2 sin^2 θ)

step 1 Question number 18, let x is equal to 2 into sine square theta, so that d x is equal to 4 into s

Determine the following: ∫0^1 √((x)/(2-x)) d x (Put .x=2...

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals containing square roots or quadratic expressions by replacing algebraic expressions with trigonometric functions. This substitution often transforms the integrand into a form involving sine, cosine, or tangent functions, where the Pythagorean identity helps in simplifying the radical expression.

Change of Variables in Definite Integrals

Changing the variable of integration through a substitution not only simplifies the integrand but also requires the transformation of the integration limits. This process ensures that the new integral is properly defined in terms of the new variable and that the solution remains consistent with the original integral's bounds.

Transformation of Integration Limits

When applying a substitution in a definite integral, the original limits must be converted to the corresponding values of the new variable. This step is crucial because it keeps the calculation self-contained and directly evaluates the integral in the new variable without having to revert back to the original variable.

Use of Trigonometric Identities

Trigonometric identities play a key role in simplifying the integrals obtained after substitution. By using identities like the Pythagorean identity, one can convert complex expressions involving square roots and fractions into simpler forms, often reducing the integral to an expression that is easier to evaluate directly.

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