$\int_1^e \frac{1}{x} \, dx$ is equal to

Let $a_{n} = \int_{-1}^{n} \left(1 + \frac{x}{2} + \frac{x^{2}}{3} + \ldots + \frac{x^{n-1}}{n}\right) dx$ for $n \in N$. Then the sum of all the elements of the set $\{n \in N : a_{n} \in (2, 30)\}$ is $...........$

Let $\lim _{c \rightarrow 0} \int_c^x \frac{b t \cos 4 t - a \sin 4 t}{t^2} d t = \frac{a \sin 4 x}{x} - 1$. Find the values of $a$ and $b$.

$3. \int_0^{\frac{1}{2}} \frac{x \sin^{-1} x}{\sqrt{1-x^2}} dx =$

$\int\limits_0^3 \left( \frac{1}{\sqrt{x^2 + 4x + 4}} + \sqrt{x^2 - 4x + 4} \right) dx =$

$\int_0^3 |x^2 - 3x + 2| dx = $