for what value(s) of m the function is increasing (Xderivative > 0)/decreasing (Xderivative < 0):

m * log(x) / 2^m + (1-x^m) / (1+x)^m

What is the pattern in the following polynom: (how to generate them) 

1:     12 n^2 + 12 n + 1

2:    300 n^3 + 450 n^2 + 160 n + 5

3:   840 n^4+1680 n^3+1030 n^2+190 n+3

4:  1260 n^5+3150 n^4+2730 n^3+945 n^2+107 n+1

5:  27720 n^6+83160 n^5+93030 n^4+47460 n^3+10689 n^2+819 n+5

1:  6n + 5

2:  150 n^2 + 200 n + 55

3:  420 n^33 + 770 n^2 + 410 n + 57

4:  630 n^4+1470 n^3+1155 n^2+343 n+29

5: 13860 n^5 + 39270 n^4 + 40740 n^3 + 18711 n^2 + 3591 n+205

int(exp(1/((x-a)*(x-b))), x = a .. b)

Let we have a function, f(x), and want to approximate its integral from x = a til x = b.

Divide the interval [a,b] into n equal (not necessary equal) parts.

Let one of the part, i,  is from x(i) til x(i+1).

Let us draw a tangent to the curve in this part, i, by the command:

Tangent(f(x), x=(x(i)+x(i+1))/2)

we may find the area under this tangent and in this interval through the command:

int(%,x=x(i)..x(i+1))

Thus in ths limit:

the original area is equal to the sum of these areas.