Integration

Ok here's something I presume is pretty easy, but I struggle. How would you integrate things like:

\frac{1+x^2}{1-x}

or

\frac{1+x}{(x^2)(k-1)}

with k constant

I don't need anyone to actually type out the whole method, just need to know what methods to use. I know they have arctan(x)'s in them, but I have no idea why :/

Thanks :smile:

try again mate, lol

OP

hmm?

Reply 3

P34c0ck1991

11

dw, it came up with error, or somthing with your latex

Reply 4

Swayum

18

First one - when dealing with algebraic fractions, you need to consider the degree of the numerator and denominator (the degree is the highest power of x). If the degree of the numerator is greater than or equal to the denominator's, then you need to carry out long division on it first. In your case, the degree of the numerator is 2 and denominator 1, and since 2 >= 1, divide it through.

Second one - how would you integrate (1 + x)/x^2?

Reply 5

c@luvr

OP

I would find the integral of

\frac{1}{x^2} + \frac{x}{x^2}

?

Reply 6

Swayum

18

Yep, exactly :smile:

.

And

\int \frac{f(x)}{k}\ \mathrm{d}x = \frac{1}{k} \int f(x)\ \mathrm{d}x

for any constant k (aside from k = 0 I guess).

Reply 7

SouthernFreerider

12

Swayum

First one - when dealing with algebraic fractions, you need to consider the degree of the numerator and denominator (the degree is the highest power of x). If the degree of the numerator is greater than or equal to the denominator's, then you need to carry out long division on it first. In your case, the degree of the numerator is 2 and denominator 1, and since 2 >= 1, divide it through.

Second one - how would you integrate (1 + x)/x^2?

clue: how could you manipulate it so that you have "(something)/x^2 + (something else)/x^2" to make it easier?