Question

Integral of 1/x^2-3 by partial fracions and by trig substituion

Answer 1

sorry its 1/x^2-2x

Answer 2

\[\frac{1}{x^2-2x}=\frac{1}{x(x-2)}=\frac{1}{2(x-2)}-\frac{1}{2x}\]

Answer 3

\[\frac{1}{x^2-2x}=\frac{1}{(x-1)^2-1}\]

Answer 4

can you show the staeps to talve that please?

Answer 5

bad answer

Answer 6

?

Answer 7

show steps and solve, im in calc II i know how to factor that demoniator on my own,,

Answer 8

the first one will just be logs and the for the second use a trig substitution.

Answer 9

recall that \[\sec^2(\theta)-1=\tan^2(\theta)\]

Answer 10

site is really slow for me today

Answer 11

i know lol can you show steps for partial fractions, idk how to find the a, b, c

Answer 12

that is the easiest part

Answer 13

\[\frac{1}{x^2-2x}=\frac{1}{x(x-2)}=\frac{A}{x-2}-\frac{B}{x}\]

\[=\frac{xA}{x(x-2)}-\frac{B(x-2)}{x(x-2)}=\frac{Ax-B(x-2)}{x(x-2)}=\frac{(A-B)x+2B}{x(x-2)}\]

so \[A-B=0\text{ and }2B=1\]

thus \[B=\frac{1}{2}\text{ and }A=\frac{1}{2}\]

Answer 14

if you used

\[\frac{1}{x^2-2x}=\frac{1}{x(x-2)}=\frac{A}{x-2}+\frac{B}{x}\]

then you would get \[B=-\frac{1}{2}\text{ and }A=\frac{1}{2}\]

but it is the same final answer

Answer 15

ok so for partial fractions i got the answer to be 1/2 ln x-2 - 1/2 ln x

Answer 16

+c

Answer 17

yes now im stuck on the trig part

Answer 18

you should really use parentheses or write in LaTeX

\[\frac{\ln|x-2|}{2}-\frac{\ln|x|}{2}+c\]

Answer 19

let \[x-1=\sec(\theta)\]

Answer 20

ok sorry, can you help me with the trig portion

Answer 21

ok i did that and got it down to sec(theta)/tan(theta) which is sin theta which has an anti derrivative of -cos theta. to me this is wrong because the answer differs

Answer 22

it does not simplify to sine

Answer 23

ok i figured that is where i went wrong, i am not sure how to solve the integral of sec(theta)/tan(theta)

Answer 24

does it simplify to csc(theta)?

Answer 25

\[\frac{\sec(\theta)}{\tan(\theta)}=\frac{1/\cos(\theta)}{\sin(\theta)/cos(\theta)}=csc(\theta)\]

Answer 26

site is sooo slow...geesh

Answer 27

yes

Answer 28

ok cool go it your the best bro!