question given ∫₂¹ f(x) dx = 11 and ∫₂⁴ f(x) ...
question given ∫₂¹ f(x) dx = 11 and ∫₂⁴ f(x) dx = -2, compute the following integral. ∫₁⁴ 6f(x) dx
Answer
# Explanation:
## Step1: Use integral property
We know that $\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx$ and $\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx$. So $\int_{1}^{4}6f(x)dx = 6\int_{1}^{4}f(x)dx$, and $\int_{1}^{4}f(x)dx=\int_{1}^{2}f(x)dx+\int_{2}^{4}f(x)dx$. Given $\int_{2}^{1}f(x)dx = 11$, then $\int_{1}^{2}f(x)dx=-\int_{2}^{1}f(x)dx=- 11$.
## Step2: Calculate $\int_{1}^{4}f(x)dx$
Substitute $\int_{1}^{2}f(x)dx=-11$ and $\int_{2}^{4}f(x)dx = - 2$ into $\int_{1}^{4}f(x)dx=\int_{1}^{2}f(x)dx+\int_{2}^{4}f(x)dx$. We get $\int_{1}^{4}f(x)dx=-11+( - 2)=-13$.
## Step3: Calculate $\int_{1}^{4}6f(x)dx$
Since $\int_{1}^{4}6f(x)dx = 6\int_{1}^{4}f(x)dx$, substitute $\int_{1}^{4}f(x)dx=-13$ into it. So $\int_{1}^{4}6f(x)dx=6\times(-13)=-78$.
# Answer:
$-78$