Integrate Using u-Substitution integral of 28(7x-2)^3 with respect to x
The problem is asking to perform an integration using the u-substitution method. This involves identifying a part of the integrand that can be substituted with a new variable, typically denoted as 'u', which simplifies the integration process. In the expression provided, the goal is to integrate a function of the form 28(7x-2)^3 with respect to x by substituting the inner function 7x-2 as 'u' to make the integral easier to evaluate.
Assign
Set
Take the derivative of
Apply the Sum Rule in differentiation to find the derivative of
Determine
Given that
Utilize the Power Rule, which states that
Multiply
Employ the Constant Rule for differentiation.
Since
Combine
Transform the integral into terms of
Extract the constant
Apply the Power Rule for integration to find the antiderivative of
Simplify the expression.
Express
Rewrite
Multiply
Substitute back the original variable:
The u-substitution method is a technique used in calculus to simplify the process of integration, particularly when dealing with composite functions. It involves the following steps:
Choosing u: Identify a part of the integrand that can be substituted with a new variable
Differentiating u: Compute
Substituting: Replace all instances of the chosen part of the integrand with
Integrating: Perform the integration with respect to
Back-substitution: Replace
Simplification: Simplify the resulting expression if possible.
The Power Rule for integration states that
The Sum Rule for differentiation states that the derivative of a sum of two functions is the sum of their derivatives:
The Constant Rule for differentiation states that the derivative of a constant is zero:
By applying these rules and the process of u-substitution, complex integrals can be evaluated more easily.