## Calculus (3rd Edition)

Given $$\int_{3}^{\infty} \frac{d x}{x^{4}+\cos ^{2} x}$$ Since for $x\geq 1$ $$\frac{1}{x^{4}+\cos ^{2} x} \geq \frac{1}{x^{4}}$$ and \begin{align*} \int_{1}^{\infty} \frac{d x}{x^{4} }&= \lim_{R\to \infty } \int_{1}^{R} \frac{d x}{x^{4} }\\ &= \lim_{R\to \infty } -\frac{1}{3x^3}\bigg|_{1}^{R} \\ &=\frac{1}{3} \end{align*} and $x^{4}+\cos ^{2} x\neq 0$, so $\int_{1}^{3} \frac{d x}{x^{4}+\cos ^{2} x}$ is finite. Hence, $\int_{3}^{\infty} \frac{d x}{x^{4}+\cos ^{2} x}$ converges