question given ∫₈⁵ f(x) dx = -5 and ∫₆⁸ f(x) ...

# Explanation: ## Step1: Use integral property We know that $\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$ and $\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx$. So, $\int_{5}^{6}f(x)dx=-\int_{6}^{5}f(x)dx$. Also, $\int_{5}^{8}f(x)dx=\int_{5}^{6}f(x)dx+\int_{6}^{8}f(x)dx$. Then $\int_{5}^{6}f(x)dx=\int_{5}^{8}f(x)dx-\int_{6}^{8}f(x)dx$. Given $\int_{8}^{5}f(x)dx = - 5$, so $\int_{5}^{8}f(x)dx=5$, and $\int_{6}^{8}f(x)dx=-9$. Then $\int_{5}^{6}f(x)dx=5-(-9)=14$. ## Step2: Use constant - multiple rule of integration The constant - multiple rule states that $\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$ for a constant $k$. Here $k = - 5$ and we want to find $\int_{5}^{6}-5f(x)dx$. So $\int_{5}^{6}-5f(x)dx=-5\int_{5}^{6}f(x)dx$. ## Step3: Substitute the value of $\int_{5}^{6}f(x)dx$ Substitute $\int_{5}^{6}f(x)dx = 14$ into the above formula. We get $\int_{5}^{6}-5f(x)dx=-5\times14=-70$. # Answer: $-70$