current objective use substitution to evaluate a definite integral with the power rule question ∫₀¹ x(3x²…

current objective use substitution to evaluate a definite integral with the power rule question ∫₀¹ x(3x² - 1)⁵ dx -determine the value of the definite integral given above. enter your answer as an exact fraction if necessary. provide your answer below:

current objective use substitution to evaluate a definite integral with the power rule question ∫₀¹ x(3x² - 1)⁵ dx -determine the value of the definite integral given above. enter your answer as an exact fraction if necessary. provide your answer below:

Answer

Explanation:

Step1: Let (u = 3x^{2}-1)

Differentiate (u) with respect to (x). We have (du=6x dx), so (x dx=\frac{1}{6}du). When (x = 0), (u=3(0)^{2}-1=-1); when (x = 1), (u=3(1)^{2}-1 = 2).

Step2: Rewrite the integral

The original integral (\int_{0}^{1}x(3x^{2}-1)^{5}dx) becomes (\frac{1}{6}\int_{-1}^{2}u^{5}du).

Step3: Apply 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, (\frac{1}{6}\int_{-1}^{2}u^{5}du=\frac{1}{6}\left[\frac{u^{6}}{6}\right]_{-1}^{2}).

Step4: Evaluate the definite integral

First, substitute the upper and lower limits: (\frac{1}{36}(2^{6}-(-1)^{6})). Then, calculate (2^{6}=64) and ((-1)^{6}=1). So, (\frac{1}{36}(64 - 1)=\frac{63}{36}=\frac{7}{4}).

Answer:

(\frac{7}{4})