current objective use substitution to evaluate a definite integral with the power rule question what is the…

Explanation:

Step1: Set the substitution

Let $u=-4x^{2}+5$. Then $du=-8x\ dx$, and $4x\ dx =-\frac{1}{2}du$.

Step2: Find new limits of integration

When $x = - 1$, $u=-4(-1)^{2}+5=-4 + 5=1$. When $x = 0$, $u=-4(0)^{2}+5 = 5$.

Step3: Rewrite the integral

The integral $\int_{-1}^{0}4x(-4x^{2}+5)^{3}dx$ becomes $-\frac{1}{2}\int_{1}^{5}u^{3}du$.

Step4: Apply the power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). So, $-\frac{1}{2}\int_{1}^{5}u^{3}du=-\frac{1}{2}\left[\frac{u^{4}}{4}\right]_{1}^{5}$.

Step5: Evaluate the definite integral

$-\frac{1}{2}\times\frac{1}{4}(u^{4})\big|_{1}^{5}=-\frac{1}{8}(5^{4}-1^{4})=-\frac{1}{8}(625 - 1)=-\frac{624}{8}=- 78$.

Answer:

$-78$