How to integrate this integral?. How to integrate $$\int\limits_{1}^{3}\cos(5x^2)\,\mathrm dx \, ?$$ Doesn't seem to work by using intergation by parts or substitution.

We can simply calculate the indefinite integral:
Substitute $u=\sqrt{\frac{10}{\pi}}x$ to get a Fresnel integral:
$$\int \cos (5x^2)\,\mathrm d x=\sqrt\frac{\pi}{10}\int\cos \left( \frac{\pi}{2}u^2\right)\,\mathrm d u=\sqrt\frac{\pi}{10} C(u)=\sqrt\frac{\pi}{10}\,C\left( \sqrt\frac{10}{\pi} x\right) +C$$
Now use the fundamental theorem of calculus to get:
$$\int_1^3 \cos (5x^2)\,\mathrm d x=\sqrt\frac{\pi}{10}\,C\left( 3\sqrt\frac{10}{\pi}\right)-\sqrt\frac{\pi}{10}\,C\left( \sqrt\frac{10}{\pi} \right)=\\
\boxed{\sqrt\frac{\pi}{10}\left( C\left(3\sqrt\frac{10}{\pi}\right)-C\left(\sqrt\frac{10}{\pi}\right)\right)}\approx 0.124$$

In general, we have:
$$ \int \cos (ax^2)\,\mathrm d x=\sqrt\frac{\pi}{2a}C\left( \sqrt\frac{2a}{\pi}x\right) + C$$
and thus
$$\int_b^c \cos (ax^2)\,\mathrm d x=\sqrt\frac{\pi}{2a}\left( C\left(c\sqrt\frac{2a}{\pi}\right)-C\left(b\sqrt\frac{2a}{\pi}\right)\right)$$
This identity can be derived analogically by substituting $u=\sqrt\frac{2a}{\pi}x$
