ca anyone pls help me out.........?
4 Answers
the integral *heavily* depends on what n is.
Case 1: n = 0.
Integral (1/(1 + 0) dx) = Integral (1)dx = x + C
Case 2: n = 1:
Integral (1 / (1 + x)) dx = ln |1 + x| + C
Case 3: n = 2:
Integral (1 / (1 + x^2)) dx = arctan(x) + C
Case 4: n = 3:
Integral (1 / (1 + x^3)) dx, and this may not even be expressable in terms of elementary functions.
How about if n = -1? Then
Integral (1 / (1 + x^(-1))dx = Integral (x / (x + 1)) dx
= Integral (1/(x + 1) + 1) dx
= ln|x + 1| + x + C
Let's see what happens if n = -2:
Integral (1 / (1 + x^(-2)) ) dx =
Integral ( (x^2) / (x^2 + 1) ) dx =
Integral ( [1 / (x^2 + 1)] + 1 ) dx =
arctan(x) + x + C
Then there are the fractional cases, like n = 1/2, 1/3, etc... which, also, may not be expressable as elementary functions.
The bottom line is that, without great difficulty, there is no way to find a general formula for the integral of (1 / (1 + x^n)).