Integrate Using u-Substitution integral of x square root of 1-x^2 with respect to x
The problem requests to perform an integration of the given function x√(1-x^2) with respect to x using the u-substitution method. U-substitution is a technique commonly used to simplify integrals by substituting a part of the integrand with a new variable, u, thus transforming the complicated integral into an easier form that can be managed with standard integration techniques. The goal is to provide an explanation of how to apply u-substitution to this integral but not to solve the integral itself.
Choose
Define
Take the derivative of
Apply the derivative operation.
Using the Sum Rule, the derivative of
The derivative of a constant is zero:
Compute
The derivative of
Apply the Power Rule, which states that the derivative of
Combine the constant with
Subtract
Transform the integral into
Begin simplifying the integral.
Place the negative sign in front of the integral:
Combine the square root and the fraction:
Extract the constant
Extract the constant
Express
Integrate
Simplify the expression.
Combine the constants and the integral result:
Simplify the fraction:
Substitute back the original
u-Substitution: A technique used in integration, which involves a change of variable to simplify the integral. It is often used when an integral contains a function and its derivative.
Derivative Rules:
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Constant Rule: The derivative of a constant is zero.
Power Rule: If
Integration Rules:
Power Rule for Integration: If
Constants can be factored out of an integral.
Square Root as a Power: The square root of a variable
Simplifying Expressions: Combining constants and simplifying fractions are common steps in finding a more compact form of the integral result.
Substituting Back: After integrating with respect to the new variable