Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Question

Is it possible to evaluate or at least to estimate the following integrals?

$$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$

and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$

I have sketchy hope to consider $\int_0^n \frac{|\sin nx| }{2+x^2+\sqrt{|\sin nx|}} dx$ and $\int_0^n \frac{|\sin nx| }{2+x+\sqrt{|\sin nx|}}$ and use symmetry somehow, but have made no progress still.

Thanks for any suggestion!

Answer 1

Denote the first integral as $f(n)$ and the second as $g(n)$.

Since $a^2/(c+a)\le 1/(c+1)$ provided $0\le a\le 1$ and $c>0$ we have that

$$f(n)\le \int_0^n \frac{1}{3+x^2} dx=\frac{\operatorname{atan}(n/\sqrt{3})}{\sqrt{3}},$$

$$g(n)\le \int_0^n \frac{1}{3+x} dx=\ln (3+n)-\ln 3.$$

Mathcad gives the following graphs.