How Do I Find the Antiderivative of x((2x^2) - 2)^3?

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Homework Help Overview

The discussion revolves around finding the antiderivative of the expression x((2x^2) - 2)^3, which falls under the subject area of calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore a substitution method by letting t = x^2 - 1, leading to a transformation of the integral into a simpler form. There are questions regarding the evaluation of the integral and the resulting values when computed over a specific interval.

Discussion Status

The discussion includes attempts to compute the integral and evaluate it over the interval from 0 to 1. Some participants express confusion regarding the results obtained, noting discrepancies between their calculations and expected outcomes.

Contextual Notes

Participants mention a potential typo in the integral expression and discuss the implications of this on their results. There is an emphasis on the evaluation limits and the resulting values from the integration process.

buffgilville
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how do I find to antiderivative of:

x((2x^2) - 2)^3
 
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buffgilville said:
how do I find to antiderivative of:

x((2x^2) - 2)^3

x(2x^2-2)^3=8x(x^2-1). Replace x^2-1 by t, notice that dt=2dx, so

\int x(2x^2-2)^3dx=\int 4 t^3 dt

then replace back

t=x^2-1

ehild
 
Last edited:
from there I was suppose to compute as x is goes from 0 to 1
I got 0, but the answer is -1
 
buffgilville said:
from there I was suppose to compute as x is goes from 0 to 1
I got 0, but the answer is -1
There was a typo in my message. It is

\int x(2x^2-2)^3dx=\int 4 t^3 dt

correctly, which is t^4= (x^2-1)^4.

\big[(x^2-1)^4\big]_0^1=(1-1)^4-(0-1)^4=-1

ehild
 

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