Question

evaluate the indefinite integral. (int x(x^{2}-3)^{3}dx). illustrate and check that your answer is reasonable by graphing both the function (f) and its antiderivative (f) (take (c = 0)).

Explanation:

Step1: Use substitution method

Let \(u = x^{2}-3\), then \(du = 2x dx\), and \(x dx=\frac{1}{2}du\).

Step2: Rewrite the integral

The integral \(\int x(x^{2}-3)^{3}dx\) becomes \(\frac{1}{2}\int u^{3}du\).

Step3: Integrate \(u^{3}\)

Using the power - rule for integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(\frac{1}{2}\times\frac{u^{4}}{4}+C=\frac{u^{4}}{8}+C\).

Step4: Substitute back \(u\)

Substituting \(u = x^{2}-3\) back, we get \(\frac{(x^{2}-3)^{4}}{8}+C\).

Answer:

\(\frac{(x^{2}-3)^{4}}{8}+C\)