Why Does This Hold True?

Question

I don't understand why this is true. Why does cos^2(x) not affect the convergence?

Answer 1

By the comparison test, we know that if $f(x)\ge g(x)\ge0$ then:

1. If $\int_{a}^{\infty}f(x)dx$ converges then so does $\int_{a}^{\infty}g (x)dx$.

2. If $\int_{a}^{\infty}g(x)dx$ diverges then so does $\int_{a}^{\infty}f(x)dx$.

Consider the first case above.

Suppose $g(x)=\cos^2(x)f(x)$. Since we know that $\cos^2(x) \in [0,1]$ we can conclude that the condition: $f(x)\ge g(x)\ge0$ is met (since $f(x)>0$ and continuous).

The proposition in the OP then follows that if $\int_{1}^{\infty}f(x)dx$ converges, then so too must $\int_{1}^{\infty}\cos^2(x)f(x)dx$.