calculate ∫5,74x²dx, given the following. ∫5,6x²dx = 91/3 ∫6,7x²dx = 127/3 ∫5,6xdx = 11/2 ∫5,74x²dx = □…

Explanation:

Step1: Use integral property

By the property $\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx$, we have $\int_{5}^{7}4x^{2}dx=4\int_{5}^{7}x^{2}dx$.

Step2: Use integral - addition property

We know that $\int_{5}^{7}x^{2}dx=\int_{5}^{6}x^{2}dx+\int_{6}^{7}x^{2}dx$.

Step3: Substitute given values

Given $\int_{5}^{6}x^{2}dx = \frac{91}{3}$ and $\int_{6}^{7}x^{2}dx=\frac{127}{3}$, then $\int_{5}^{7}x^{2}dx=\frac{91}{3}+\frac{127}{3}=\frac{91 + 127}{3}=\frac{218}{3}$.

Step4: Calculate the final result

Since $\int_{5}^{7}4x^{2}dx=4\int_{5}^{7}x^{2}dx$, substituting $\int_{5}^{7}x^{2}dx=\frac{218}{3}$, we get $4\times\frac{218}{3}=\frac{872}{3}$.

Answer:

$\frac{872}{3}$