What is the integral of 3(x^2 - 2)^4 dx using substitution?

To compute the integral of ( 3(x^2 - 2)^4 , dx ) using substitution, we can start by making an appropriate substitution. Let ( u = x^2 - 2 ). Then, the differential ( du ) is given by:

[

du = 2x , dx \quad \Rightarrow \quad dx = \frac{du}{2x}

]

However, we will need to express ( x ) in terms of ( u ) to substitute ( dx ) correctly. Since ( u = x^2 - 2 ), we have ( x^2 = u + 2 ) and thus ( x = \sqrt{u + 2} ) or ( x = -\sqrt{u + 2} ). However, in most integrals involving ( x^2 ), we can just keep ( x ) symbolic since ( x ) will cancel out later.

Now, substitute ( u ) back into the integral:

[

\int 3(x^2 - 2)^4 , dx = \int 3u^4 , \left(\frac{du}{2x}\right)