4 digits

Sam fisher have accomplished a mission. But while leaving the area, he forgot the 4 digit door code. He can't open the door without the 4 digit code.

However, he got some clues:

• The first digit is 2.

• The second digit is $\displaystyle \int^2_0 \dfrac 24 - \dfrac 1 2 dx$

• The third digit is $\displaystyle \lim_{x \to \infty} \dfrac{2-x}{4-x}$

• The fourth digit is $\large 2^{2^0} + 6$

What is the 4 digit integer?