which of the following definite integrals is equal to lim(n→∞)∑(k = 1)^n (12k/n)cos(1 + 4k/n)(4/n)?\na ∫_1^5…

which of the following definite integrals is equal to lim(n→∞)∑(k = 1)^n (12k/n)cos(1 + 4k/n)(4/n)?\na ∫_1^5 12cosxdx\nb ∫_1^5 3xcosxdx\nc ∫_0^4 12cos(1 + x)dx\nd ∫_0^4 3xcos(1 + x)dx

which of the following definite integrals is equal to lim(n→∞)∑(k = 1)^n (12k/n)cos(1 + 4k/n)(4/n)?\na ∫_1^5 12cosxdx\nb ∫_1^5 3xcosxdx\nc ∫_0^4 12cos(1 + x)dx\nd ∫_0^4 3xcos(1 + x)dx

Answer

Explanation:

Step1: Recall the definition of definite integral

The definite - integral $\int_{a}^{b}f(x)dx=\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}f(x_k)\Delta x$, where $\Delta x=\frac{b - a}{n}$ and $x_k=a + k\Delta x$. Given $\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\frac{12k}{n}\cos(1+\frac{4k}{n})\frac{4}{n}$, we have $\Delta x=\frac{4}{n}$. Then, by $\Delta x=\frac{b - a}{n}$, we get $b - a = 4$.

Step2: Find $x_k$ and $f(x)$

We have $x_k=1+\frac{4k}{n}$, so $a = 1$ and $b=a + 4=5$. Also, $\frac{12k}{n}\cos(1+\frac{4k}{n})\frac{4}{n}=3\times(1+\frac{4k}{n})\cos(1+\frac{4k}{n})\frac{4}{n}$. Let $x = 1+\frac{4k}{n}$, then $f(x)=3x\cos x$.

Answer:

B. $\int_{1}^{5}3x\cos xdx$