Question

the indefinite integral can be found in more than one way. first use the substitution method to find the indefinite integral. then find it without substitution. check that your answers are equivalent.

$\int 3x^{2}(x^{3}-3)dx$

use the substitution method to find the indefinite integral. what expression should be used for the substitution, u?

a. 3x
b. $x^{3}-3$
c. x
d. $x^{3}$

Explanation:

Step1: Recall substitution rule

If \(u = x^{3}-3\), then \(du=3x^{2}dx\). The integral \(\int3x^{2}(x^{3} - 3)dx\) can be rewritten in terms of \(u\).

Step2: Substitute \(u\) and \(du\)

Substituting \(u\) and \(du\) into the integral \(\int3x^{2}(x^{3}-3)dx=\int udu\).

Answer:

B. \(x^{3}-3\)