All Tasks - Asymptote

Consider the integral $\displaystyle I=\int_{0}^{a} 2\sqrt{a^2-x^2}dx$ and the change of variable $x=a \sin(t)$. So $I$ comes equal to:

Consider the integral $\displaystyle I=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} f(x)dx$ and the change of variable $x= \arctan(t)$. So $I$ comes equal to:

The value of the integral $\int_1^{e}\ln(x)dx$ is:

The value of the integral $\displaystyle \int_0^1 \frac{y}{y^2+1}dy$ is:

If $f$ is a real function of a real variable, is continues and $F'(x)=f(x), \forall x \in \mathbb{R}$, so $\int_1^3 f(2x)dx$ is:

Imagine that you want to buy some pens. In the shop near home one pen cost 0,80 euro, online one pen cost 0,60 euro but the shipping fees are 3 euro. How many pens you have to buy in order to spend the same money in each shop?