how do we split this up for integrating ?

Question

aravindg · Answer

$$\large \int\limits \dfrac{x^3+x+1}{x^2-1}dx$$

aravindg · Answer

we cant apply partial fraction directly isnt it ?

callisto · Answer

First, do long division, then, partial fraction.

ash2326 · Answer

Divide the numerator by denominator first, then proceed
$$\int (x+\frac{2x+1}{x^2-1} )dx$$
$$\int (x+\frac {2x}{x^2+1}+\frac{1}{x^2-1} )dx$$
Straightforward now

callisto · Answer

Since the degree of polynomial in numerator > that in denominator, you can't use partial fraction directly.

ash2326 · Answer

oops
$$\int (x+\frac {2x}{x^2-1}+\frac{1}{x^2-1} )dx$$

aravindg · Answer

I am having a difficulty in dividing

ash2326 · Answer

By splitting, I have inadvertently used partial fractions. I've not shown that explicitly

aravindg · Answer

what if both the numerator and denominator are of same degree ? What do I do then ?

ash2326 · Answer

You have to divide then too :)

ash2326 · Answer

you have x^3+x+1 / x^2-1
x^3/x^2= x
so
the first term is x, the extra term is -x, you add +x  to the remaining
you get x as quotient and 2x+1 as remainder

aravindg · Answer

like this ? 

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