Convergence and Divergence and Using Various Methods

Question

I am totally confused with the idea of convergence and divergence and which method to use to proof it.

An example is a question like this:

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Does this integral converge? $$\int_{12}^\infty x^{-x}\mathsf dx $$

Another question is

Does this integral converge? $$\int_1^\infty \left(\frac{\sin x}x\right)^2\mathsf dx$$

Any help will be greatly appreciated.

I know the various theorems such as squeeze, integral, p-test but just need a bit of help.

Answer 1

Some hints:

$$\int_{12}^\infty \frac{1}{x^x}\, dx \le \int_{12}^\infty \frac{1}{x^{12}}\,dx.$$

When thinking about $\int_1^\infty (\sin^2x/x^2)\,dx,$ what is the simplest bound on $\sin^2 x$ you can think of?