Evaluate `int (x-1)/(x^2-2x+1)dx`

Answer

4

Explanation

a. `underset(r=0)overset(14)sum((15-r))/(r+1)(5!)/(r!(15-r)!).^(20)C_(m-r)`
`= underset(r=0)overset(14)sum((15-r))/(r+1)(15!)/(r!(15-r)!).^(20)C_(m-r)`
`= underset(r=0)overset(14)sum(15!)/((r+1)!(14-r)!) .^(20)C_(m-r)`
`= underset(r=0)overset(14)sum.^(15)C_(r+1).^(20)C_(m-r)`
`= .^(35)C_(m+1)`.
Which has maximum value `.^(35)C_(18)`.
b. `.^(35)C_(0)-.^(35)C_(1)+.^(35)C_(2) - .^(35)C_(3)+"....."+(-1)^(r ) .^(35)C_(r) + "......"`
= Coefficient of `x^(r)` in
`(.^(35)C_(0)-.^(35)C_(1)x+.^(35)C_(2)x^(2)-.^(35)C_(3)x^(3)+"...."+(-1)^(r).^(35)C_(r)+"....")xx(1+x+x^(2)+x^(3)+"....."+x^(r)+".....")`
= Coefficient of `x^(r)` in `(1-x)^(35)(1-x)^(-1)`
= Coefficient of `x^(r)` in `(1-x)^(34)`
`= (-1)^(r) .^(34)C_(r)`
Required maximum value is `.^(24)C_(17)`.
c. `.^(31)C_(r-5)+5xx.^(31)C_(r-4)+10xx.^(31)C_(r-3)+10xx.^(31)C_(r-2)+5xx.^(31)C_(r-1)+.^(31)C_(r)`
`= .^(5)C_(5)xx.^(31)C_(r-5)+.^(5)C_(4)xx.^(31)C_(r-4)+.^(5)C_(3)xx.^(31)C_(r-3)+.^(5)C_(2)xx.^(31)C_(r-2) +.^(5)C_(1)xx.^(31)C_(r-1)+.^(5)C_(0)xx.^(31)C_(r)`
`=` Coefficient of `x^(r)` in `(1+x)^(5)(1+x)^(31)`
= Coefficient of `x^(r)` in `(1+x)^(36)`
`= .^(36)C_(r)`
Which has maximum value of `.^(36)C_(18)`.
d. `underset(r=0)overset(k)sum2^(k-r).(-1)^(r).^(37)C_(r)..^(37-r)C_(37-k)`
`= underset(r=0)overset(k)sum2^(k-r).(-1)^(r).(37!)/((37-r)!r!)((37-r)!)/((37-k)!(k-r)!)`
`= (37!)/(k!(37-k)!) underset(r=0)overset(k)sum2^(k-r).(-1)^(r).(k!)/(r!(k-r)!)`
`= .^(37)C_(r)underset(r=0)overset(k)sum.^(k)C_(r)2^(k-r).(-1)^(r)`
`= .^(37)C_(k)(2-1)^(k)`
`= .^(37)C_(k)`