current objective use substitution to evaluate a definite integral with the power rule question ∫ - 1 0 - 5…

Explanation:

Step1: Let (u = x^{2}-1)

Differentiate (u) with respect to (x): (du=2x dx), so (x dx=\frac{1}{2}du). When (x = - 1), (u=(-1)^{2}-1 = 0); when (x = 0), (u=0^{2}-1=-1). The integral (\int_{-1}^{0}-5x(x^{2}-1)^{4}dx) becomes (\int_{0}^{-1}-5\times\frac{1}{2}u^{4}du=\frac{5}{2}\int_{-1}^{0}u^{4}du).

Step2: Apply the power - rule for integration

The power - rule for integration is (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). For (\int u^{4}du=\frac{u^{5}}{5}+C). Then (\frac{5}{2}\int_{-1}^{0}u^{4}du=\frac{5}{2}\left[\frac{u^{5}}{5}\right]_{-1}^{0}).

Step3: Evaluate the definite integral

First, substitute the upper and lower limits: (\frac{5}{2}\times\frac{1}{5}(0^{5}-(-1)^{5})=\frac{1}{2}(0 + 1)=\frac{1}{2}).

Answer:

(\frac{1}{2})