Integrate Using u-Substitution integral of x/( square root of 1-x^2) with respect to x
The problem is asking for the evaluation of an integral using the method of u-substitution. The integral in question is the integral of the function x/(√(1-x^2)) with respect to x. To approach this integral with u-substitution, you typically look for a function inside the integral that, when differentiated, appears elsewhere in the integral. The integral suggests a potential substitution with u being related to the expression (1-x^2) because its derivative, -2x, is related to the x in the numerator. The goal is to rewrite the integral in terms of u to simplify the integration process. The u-substitution method is a common technique used to evaluate more complex integrals by changing variables to simplify the integral before performing the integration.
Choose
Set
Take the derivative of
Proceed with differentiation.
Using the Sum Rule, the derivative of
The derivative of a constant,
Evaluate the derivative of
Applying the constant multiple rule, the derivative of
Apply the Power Rule, which states that the derivative of
Combine the constant
Subtract
Express the integral in terms of
Begin simplification.
Extract the negative sign from the fraction:
Combine
Rearrange the fraction:
Extract the constant
Extract the constant
Apply exponent rules.
Express
Rewrite
Simplify the exponent.
Apply the power of a power rule:
Combine the exponents:
Keep the negative sign outside the fraction:
Integrate using the Power Rule:
Final simplification.
Multiply
Combine terms:
Substitute back
The u-substitution method is a technique used in calculus to simplify the process of integration by substituting a part of the integrand with a new variable, typically denoted as
The steps involved in u-substitution generally include:
Identifying a part of the integrand to substitute with
Differentiating
Substituting
Simplifying the integral if possible.
Integrating with respect to
Substituting back the original variable
The Power Rule for integration states that the integral of
The Sum Rule for differentiation states that the derivative of a sum of functions is the sum of their derivatives.
The Constant Multiple Rule allows us to take constants out of the derivative or integral.
The Power Rule for differentiation states that the derivative of
These rules are fundamental to calculus and are used extensively in solving integration and differentiation problems.