determine whether the following integral converges or diverges.\n∫1,∞ (8 + cos 2x)/x^7 dx\nas x approaches…

determine whether the following integral converges or diverges.\n∫1,∞ (8 + cos 2x)/x^7 dx\nas x approaches ∞, which of the following is true about the behavior of the integrand (8 + cos 2x)/x^7?\na. 0 ≤ 9/x^7 ≤ (8 + cos 2x)/x^7\nb. 0 ≤ (8 + cos 2x)/x^7 ≤ 9/x^7\nc. 0 ≤ (8 + cos 2x)/x^7 ≤ cos 2x/x^7\nd. 0 ≤ 8/x^7 ≤ (8 + cos 2x)/x^7

determine whether the following integral converges or diverges.\n∫1,∞ (8 + cos 2x)/x^7 dx\nas x approaches ∞, which of the following is true about the behavior of the integrand (8 + cos 2x)/x^7?\na. 0 ≤ 9/x^7 ≤ (8 + cos 2x)/x^7\nb. 0 ≤ (8 + cos 2x)/x^7 ≤ 9/x^7\nc. 0 ≤ (8 + cos 2x)/x^7 ≤ cos 2x/x^7\nd. 0 ≤ 8/x^7 ≤ (8 + cos 2x)/x^7

Answer

Explanation:

Step1: Recall the range of cosine function

The range of $\cos(2x)$ is $[- 1,1]$. So, $8+\cos(2x)$ has a range. When $\cos(2x)=1$, $8 + \cos(2x)=9$; when $\cos(2x)=-1$, $8+\cos(2x)=7$.

Step2: Determine the inequality for the integrand

Since $7\leqslant8 + \cos(2x)\leqslant9$ and $x>0$ (in the context of the integral $\int_{1}^{\infty}\frac{8+\cos(2x)}{x^{7}}dx$), then $0\leqslant\frac{8 + \cos(2x)}{x^{7}}\leqslant\frac{9}{x^{7}}$.

Answer:

B. $0\leqslant\frac{8+\cos(2x)}{x^{7}}\leqslant\frac{9}{x^{7}}$