current objective use substitution to evaluate a definite integral with the power rule question find ∫ - 1 0…

Explanation:

Step1: Let (u = 3x^{2}-2)

Differentiate (u) with respect to (x): (du=6x dx), so (- 3x dx=-\frac{1}{2}du). When (x=-1), (u = 3\times(-1)^{2}-2=1). When (x = 0), (u=3\times0^{2}-2=-2).

Step2: Rewrite the integral

The integral (\int_{-1}^{0}-3x(3x^{2}-2)^{4}dx) becomes (\int_{1}^{-2}-\frac{1}{2}u^{4}du).

Step3: Use the power - rule for integration

The power - rule for integration is (\int u^{n}du=\frac{u^{n + 1}}{n+1}+C) ((n\neq - 1)). So (\int-\frac{1}{2}u^{4}du=-\frac{1}{2}\times\frac{u^{5}}{5}+C=-\frac{u^{5}}{10}+C).

Step4: Evaluate the definite integral

(\int_{1}^{-2}-\frac{1}{2}u^{4}du=\left[-\frac{u^{5}}{10}\right]_{1}^{-2}=-\frac{(-2)^{5}}{10}-\left(-\frac{1^{5}}{10}\right)). [ \begin{align*} &=-\frac{-32}{10}+\frac{1}{10}\ &=\frac{32 + 1}{10}\ &=\frac{33}{10} \end{align*} ]

Answer:

(\frac{33}{10})