calculate the integral, assuming that $\\int_{0}^{5} f(x) d x=2$ and $\\int_{0}^{5} g(x) d x=15$. (give your…
calculate the integral, assuming that $\\int_{0}^{5} f(x) d x=2$ and $\\int_{0}^{5} g(x) d x=15$. (give your answer as a whole or exact number.) $\\int_{0}^{5}\\left(6 f(x)+\\frac{1}{3} g(x)\\right) d x=$
Answer
Explanation:
Step1: Use integral linearity property
$$\int_{0}^{5}\left(6f(x)+\frac{1}{3}g(x)\right)dx = 6\int_{0}^{5}f(x)dx+\frac{1}{3}\int_{0}^{5}g(x)dx$$
Step2: Substitute the given values
Given $\int_{0}^{5}f(x)dx = 2$ and $\int_{0}^{5}g(x)dx = 15$. Substitute into the above formula: $$6\times2+\frac{1}{3}\times15$$
Step3: Calculate the result
First calculate $6\times2 = 12$, then calculate $\frac{1}{3}\times15=5$. Then $12 + 5=17$.
Answer:
17