(6) $\\int_{0}^{1}2(x^{2}-1)^{100}xdx$

Explanation:

Step1: Use substitution

Let (u = x^{2}-1), then (du = 2xdx). When (x = 0), (u=-1); when (x = 1), (u = 0). The integral (\int_{0}^{1}2(x^{2}-1)^{100}xdx=\int_{-1}^{0}u^{100}du)

Step2: Integrate (u^{n})

According to the power - rule of integration (\int u^{n}du=\frac{u^{n + 1}}{n+1}+C(n\neq - 1)). Here (n = 100), so (\int_{-1}^{0}u^{100}du=\left[\frac{u^{101}}{101}\right]_{-1}^{0})

Step3: Evaluate the definite integral

(\left[\frac{u^{101}}{101}\right]_{-1}^{0}=\frac{0^{101}}{101}-\frac{(-1)^{101}}{101}) Since (0^{101}=0) and ((-1)^{101}=-1), then (\frac{0^{101}}{101}-\frac{(-1)^{101}}{101}=0+\frac{1}{101})

Answer:

(\frac{1}{101})