All Tasks - Asymptote

Considere o integral $\displaystyle I=\int_{0}^{a} 2\sqrt{a^2-x^2}dx$ e a mudança de variável $x=a\sin(t)$. Então I vem igual a:

Considere o integral $\displaystyle I=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} f(x)dx$ e a mudança de variável $x= \arctan( t)$. Então $I$ é igual a:

O valor do integral $\int_1^{e}\ln(x)dx$ é:

O valor do integral $\displaystyle \int_0^1 \frac{y}{y^2+1}dy$ é:

Se $f$ é uma função real de variável real, continua e $F'(x)=f(x), \forall x \in \mathbb{R}$, então $\int_1^3 f(2x)dx$ é:

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?