(Hint. Re-arrange the Pascals triangle in the following
way:

`((1, , , , , , ,),(1, 1, , , , , , ),(1, 2,bb1, , , , , ),(1, bb3,3,1,
, , , ),(bb1, 4,6,4,1, , , ),(1, 5,10,10,5,1, , ),(1, 6,15,20,15,6,1, ),(1,
7,21,35,35,21,7,1))`

For each row, add up the first number of the row, the
second number of its first upper row,
the third number of its second upper row,
and so on until there are no numbers to add up.
For example, `bb1+bb3+bb1=5`
for the 5th row and `1+4+3=8` for the 6th row.)

`1,1,2,3,5,8,13,21,34,55,89,cdots`

If `A=>B` is true, then its contrapositive (`notB=>notA`) is always
true
and if `A=>B` is false, then `notB=>notA` is always false,
but its
converse (`B=>A`) or inverse (`notA=>notB`) is not always so.

Prove this
theorem using a truth table.

`int sqrt(1+4x^2)dx =
1/2xsqrt(1+4x^2)+1/4ln|2x+sqrt(1+4x^2)| + C`

`int sqrt(1+x^4)/(x^3)dx = -
1/2sqrt(1+x^4)/(x^2) + 1/2ln|x^2+sqrt(1+x^4)| + C`

(Note that the first one
is a counter part of the famous `int sqrt(1-4x^2)dx`
in this forum and the
second one is needed to calculate
the surface area of Gabriels horn.)