Consider the integral ∫x(x^2-1)^2 d x. (a) Solve this integral by using u -substitution. (b) Sol

step 1 Hello everyone, so in this question the integral given to us is x squared multiplied by l n x cu

Consider the integral ∫x(x^2-1)^2 d x. (a) Solve this...

Consider the integral $\int x\left(x^{2}-1\right)^{2} d x$.
(a) Solve this integral by using $u$ -substitution.
(b) Solve the integral another way, using algebra to multiply out the integrand first.
(c) How must your two answers be related? Use algebra to prove this relationship.

Key Concepts

Constant of Integration

The Constant of Integration is a critical concept in the context of indefinite integrals. Since indefinite integration represents the antiderivative of a function, it is not unique and includes an arbitrary constant. Different methods of integration may yield antiderivatives that differ by a constant, yet they are equivalent representations of the same family of functions.

Equivalence of Antiderivatives

The Equivalence of Antiderivatives refers to the fact that any two antiderivatives of the same function differ only by a constant. When solving an integral using different techniques, verifying that the results are equivalent (up to an additive constant) confirms the correctness of the methods applied and is fundamental to understanding the nature of integration.

u-Substitution

u-Substitution is a fundamental integration technique used to simplify integrals by making a substitution that transforms a composite function into a simpler form. By letting u equal an inner function of the integrand, the differential dx is expressed in terms of du, which often turns the integral into a basic form that can be integrated directly.

Algebraic Expansion

Algebraic Expansion involves multiplying out or simplifying an integrand before integration. This method is particularly useful when the integrand is given in a factored or compounded form. By expanding the expression, one can split the integral into a sum of integrals of simpler terms, each of which may be integrated using standard rules.

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