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

In summary, to find the antiderivative of x((2x^2) - 2)^3, replace x^2-1 with t and dt=2dx. Then, substitute t=x^2-1 and compute from 0 to 1 to get the answer of -1.
  • #1
buffgilville
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0
how do I find to antiderivative of:

x((2x^2) - 2)^3
 
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  • #2
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

[tex]\int x(2x^2-2)^3dx=\int 4 t^3 dt [/tex]

then replace back

[tex]t=x^2-1[/tex]

ehild
 
Last edited:
  • #3
from there I was suppose to compute as x is goes from 0 to 1
I got 0, but the answer is -1
 
  • #4
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

[tex]\int x(2x^2-2)^3dx=\int 4 t^3 dt [/tex]

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

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

ehild
 

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

What is the antiderivative of X((2x^2) - 2)^3?

The antiderivative of X((2x^2) - 2)^3 is (1/5)x^5 - (2/3)x^3 + (2/3)x + C.

How do you solve for the antiderivative of X((2x^2) - 2)^3?

To solve for the antiderivative of X((2x^2) - 2)^3, you can use the power rule and the chain rule to integrate each term separately. Then, you can add the constant of integration (C) to the end of the expression.

Can the antiderivative of X((2x^2) - 2)^3 be simplified?

Yes, the antiderivative of X((2x^2) - 2)^3 can be simplified using algebraic techniques. However, it is generally recommended to leave the expression in its expanded form to avoid confusion.

Is there a shortcut to finding the antiderivative of X((2x^2) - 2)^3?

No, there is no specific shortcut for finding the antiderivative of X((2x^2) - 2)^3. However, familiarizing yourself with integration techniques and practicing regularly can make the process faster and more efficient.

Can the antiderivative of X((2x^2) - 2)^3 be used to find the definite integral?

Yes, the antiderivative of X((2x^2) - 2)^3 can be used to find the definite integral by simply plugging in the upper and lower limits of integration and subtracting the results. This is known as the fundamental theorem of calculus.

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