Continuity and integration relationship

Question

Is there a relation between the integral of a function and its continuity? For example, $\sin x^2$ is continuous but not integrable.

Answer 1

$$x\mapsto sin (x^2)$$ is continuous at $\Bbb R $, thus it is integrable at any compact $[a,x] $.

Its integrale function is defined by $$F : x\mapsto \int_0^x \sin (t^2)dt $$

it can't be expressed easily. You can call it for example $$Sin_2 (x). $$

Like $\int_1^x\frac {dt}{t} $ is called $\ln (x) $.