Question

step 1
we know that if ( u = f(x) ), then ( du = f(x) dx ). therefore, if ( u = x^{3}+38 ), then ( du = 3x^{2} dx ).
step 2
if ( u = x^{3}+38 ) is substituted into ( int x^{2}sqrt{x^{3}+38} dx ), then we have ( int x^{2}(u)^{1/2} dx=int u^{1/2}x^{2} dx ).
we must also convert ( x^{2} dx ) into an expression involving ( u ).
we know that ( du = 3x^{2} dx ), and so ( x^{2} dx = du ).

Explanation:

Step1: Solve for \(x^{2}dx\) in terms of \(du\)

Given \(du = 3x^{2}dx\).
Divide both sides by \(3\):
\(x^{2}dx=\frac{1}{3}du\)

Answer:

\(\frac{1}{3}\)