Question

use the substitution method to find the indefinite integral. what expression should be used for the substitution, u?
a. 3x
b. x³ - 3
c. x
d. x³
find the indefinite integral.
∫3x²(x³ - 3)dx =
∫3x²(x³ - 3)dx =
find the indefinite integral without using substitution.
∫3x²(x³ - 3)dx = □

Explanation:

Step1: Recall substitution rule

If \(u = x^{3}-3\), then \(du=3x^{2}dx\).

Step2: Apply substitution

\(\int3x^{2}(x^{3} - 3)dx=\int udu\).

Step3: Integrate \(\int udu\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), for \(n = 1\), \(\int udu=\frac{u^{2}}{2}+C\).

Step4: Substitute back \(u=x^{3}-3\)

\(\frac{(x^{3}-3)^{2}}{2}+C\).

Answer:

\(\frac{1}{2}(x^{3}-3)^{2}+C\)