Integrate (1+x+x^2) / x(1+x^2) for x. Partial dractions?

[teng125]

integre (1+x+x^2) / x(1+x^2)
pls help...
i used partial fraction and obtain the answer of ln x - 1/2 ln (1-x^2) + arctan x
is it correct??

 

[HallsofIvy]

That's not what I get. HOW did you get that? What was the partial fractions expansion?

 

[teng125]

i got 1 +x +x^2 = A(1+x^2) + B(x^2) + Cx
then let x =1 and -1 and again x=0

is it correct??

 

[VietDao29]

Looks good. So what are your A, B, and C?
Remember that:
$$\frac{1 + x + x ^ 2}{x(1 + x ^ 2)} = \frac{A}{x} + \frac{Bx + C}{1 + x ^ 2}$$
What does your final result look like?

 

[teng125]

A=1 B=0 and C=1

 

[teng125]

is it right

 

[VietDao29]

Yup, it looks good. So:
$$\int \frac{1 + x + x ^ 2}{x(1 + x ^ 2)} dx = \int \left( \frac{1}{x} + \frac{1}{1 + x ^ 2} \right) dx = ?$$
Can you integrate this? And what's your answer, then?

 

[teng125]

oh...i intgre wrongly on the above...is should be ln x + arctan x rite??

 

[VietDao29]

Looks about right, but don't forget the constant of integration.

 

[Ouabache]

Are they teaching integration in precalculus now?