thanks mathgirl8, here's another 10 pts. what is the integral of (x^3+1)/(x^2+1) dx?

Answer 1

Use long division to simplify.
(x^3 +1)/(x^2 +1) = x + (-x+1)/(x^2 +1)
Separate the fraction into two parts
= x - x/(x^2 +1) + 1/(x^2 +1)
The integral of x is (1/2)x^2
The integral of 1/(x^2 +1) = arctan(x)
To integrate x/(x^2 +1), you can use substitution. Let u = x^2 +1
Answer: (1/2)x^2 - (1/2)ln(x^2 +1) + arctan(x) + C

Answer 2

x^3 + 1 = (x^2 + 1) * x + (1 - x), so the integral splits as

INT x dx + INT 1/(x^2 + 1) dx - INT x/(x^2 + 1) dx

The first term becomes 1/2 x^2.
The second term becomes inv tan x (or arc tan x).
The third term becomes -1/2 ln |x^2 + 1|.

Conclusion:
1/2 x^2 + inv tan x - 1/2 ln |x^2 + 1| + C.

Answer 3

x^5+x^3+x^2+1 dx and use quadratic formula