Question

another one i got wrong...

\[\large \int_5^{\infty} \frac{2dx}{x\sqrt{x^2 - 25}}\]

Answer 1

my answer was \(\large \frac{\pi}{5}\) and got 6 points out of 12 =_=

Answer 2

@myininaya ...need a consult

Answer 3

\[\int\limits_{}^{}\frac{2 dx}{x \sqrt{x^2-25}}\]
Did you have a problem integrating?

Answer 4

uhmm you can judge for yourself... \[2\lim_{t \rightarrow \infty} \int_5^t \frac{dx}{x\sqrt{x^2 - 25}}\]

Answer 5

\[2[\frac{1}{5} \sec^{-1} (\frac{x}{5})|_5^t]\]

Answer 6

\[2[\frac{1}{5} \sec^{-1} (\infty)] - 2[\frac{1}{5} \sec^{-1} (1)]\]

Answer 7

\[2[\frac{1}{5} (\frac{\pi}{2})] - 2[\frac{1}{5} (0)]\]

Answer 8

\[\int\limits \frac{2 dx}{x \sqrt{x^2-25}}\]

I labeled it this way so that i would have what is inside that square rooty :)
\[\sec(\theta)=\frac{x}{5} => \sec(\theta) \tan(\theta) d \theta=\frac{1}{5} dx\]
So we have
\[2 \int\limits_{}^{}\frac{5 \sec(\theta) \tan(\theta) d \theta}{5 \sec(\theta) \sqrt{(5\sec(\theta))^2-25}}\]
\[\frac{2}{5}\int\limits_{}^{}\frac{\sec(\theta) \tan(\theta) d \theta}{\sec(\theta)\sqrt{\sec^2(\theta)-1}}\]
\[\frac{2}{5}\int\limits_{}^{}1 d \theta=\frac{2}{5}\theta+C=\frac{2}{5}\sec^{-1}(\frac{x}{5})+C\]
Ok I approved of your integrating part

Answer 9

lol

Answer 10

lol :p so where did i go wrong

Answer 11

\[\int\limits_{5}^{52} f(x) dx+\int\limits_{52}^{\infty}f(x) dx\]
I just split them up at some random constant between 5 and infinity lol
\[[\lim_{a \rightarrow 5^+}\frac{2}{5}\sec^{-1}(\frac{x}{5})]_a^{52}+[\lim_{b \rightarrow \infty}\frac{2}{5} \sec^{-1}(\frac{x}{5})]_{52}^{b}\]

Answer 12

sorry it keeps going to that aw snap page so annoying

Answer 13

it is isnt it

Answer 14

\[\lim_{a \rightarrow 5^+}\frac{2}{5}(\sec^{-1}(\frac{52}{5})-\sec^{-1}(\frac{a}{5}))+\lim_{b \rightarrow \infty} \frac{2}{5}(\sec^{-1}(\frac{b}{5})-\sec^{-1}(\frac{52}{5}))\]
Now do both parts converge?

Answer 15

by converge means definite value right?

Answer 16

seems like it

Answer 17

a finite value yes and yes you are right

Answer 18

lol

Answer 19

\[\frac{2}{5}(\sec^{-1}(\frac{52}{5})-0)+\frac{2}{5}(\frac{\pi}{2}-\sec^{-1}(\frac{52}{5}))\]
\[\frac{2}{5} \cdot \frac{\pi}{2}\]

Answer 20

pi/5 is right

Answer 21

then why did i get 6 out of 12 points =_=

Answer 22

maybe she/he didn't like the way that you didn't split it up

Answer 23

oh lol

Answer 24

there was 2 different discont.\ infinite situation going on there

Answer 25

finkledipper -_-

Answer 26

so you need to do 2 separate thingys lol

Answer 27

well thanks :D

Answer 28

you're welcome

Answer 29

by the way if you haven't noticed i don't like using the formulas
i like do it the long way every time lol

Answer 30

haha yeah i noticed :p i did that too but too much latex and drawings so i just wrote it directly hehe