Question

Made a calculus question for fun. \[\int\frac{5 \cos(x)-2 \sin(x)}{ 5 \sin(x)-2 \cos(x)} dx\]

Answer 1

By the way I'm not sure if my way is the hard way or not but I like it and I think it is fun.

Answer 2

I'm starting with \[a\cos x +b\sin x= C\cos(x +\phi)\]

Answer 3

@ganeshie8 can you help me with a question please :)

Answer 4

err that is my way @ganeshie8 :p

Answer 5

So problem is too easy. :p

Answer 6

someone please help :(

Answer 7

@ganeshie8 ? @freckles ?

Answer 8

You could mention your question in chat @geny55 .

Answer 9

is moisture supply abiotic or biotic?

Answer 10

Well this isn't really chat... but okay.

Answer 11

lol

Answer 12

so ??

Answer 13

I don't know.

Answer 14

You might mention your question though in chat by posting a link to your question .

Answer 15

\[\frac{5 \cos(x)-2 \sin(x)}{ 5 \sin(x)-2 \cos(x)} =\dfrac{\cos(x - \arctan(-2/5))}{
\cos(x-\arctan(-5/2))}\]

?

Answer 16

idk readily what to next..

Answer 17

@ganeshie8 is moisture supply abiotic or biotic?

Answer 18

\[\frac{\cos(a+x)}{\cos(b+x)} =\frac{\cos(x+b+(a-b))}{\cos(b+x)}\]

Answer 19

use sum identity on top

Answer 20

Wow! that simplifies nicely !

Answer 21

Let \(a = \arctan(-2/5)\) and \(b = \arctan(-5/2)\)

\[\begin{align} &\frac{5 \cos(x)-2 \sin(x)}{ 5 \sin(x)-2 \cos(x)}\\~\\
& =\dfrac{\cos(x -a)}{\cos(x-b)}\\~\\
& =\dfrac{\cos((x -b)+(b-a))}{\cos(x-b)}\\~\\
&=\cos(b-a) - \sin(b-a)\tan(x-b)
\end{align}\]

Answer 22

\[\int\limits \frac{ a \cos(x)+b \sin(x)}{c \cos(x)+d \sin(x)} dx \\ \frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}\int\limits \frac{\cos(x+A)}{\cos(x+B)} dx \\ \sqrt{\frac{a^2+b^2}{c^2+d^2}} \int\limits \frac{\cos(x+B+(A-B)}{\cos(x+B)} dx \\ \sqrt{\frac{a^2+b^2}{c^2+d^2}} \int\limits \frac{\cos(x+B) \cos(A-B) -\sin(x+B)\sin(A-B)}{\cos(x+B)} dx \\ \sqrt{\frac{a^2+b^2}{c^2+d^2}} ( \int\limits \cos(A-B) dx-\sin(A-B) \int\limits \tan(x+B) dx) \\ \sqrt{\frac{a^2+b^2}{c^2+d^2} } (\cos(A-B)x+\sin(A-B) \ln|\cos(x+B)|) +K\]
if I didn't make a mistake we can do a more general form in this way

Answer 23

\[\sqrt{ \frac{a^2+b^2}{c^2+d^2}} \cos(A-B) x+\sqrt{\frac{a^2+b^2}{c^2+d^2}}\sin(A-B) \ln|\cos(x+B)|+K \\ \text{ where } A=\arctan(\frac{b}{a}) \text{ and } B=\arctan(\frac{d}{c})\]

Answer 24

there are some restrictions on a,b,c,d

Answer 25

Nice! that wasn't so hard as it appeared to be..

Answer 26

There are only ways people presented here
but I was very much in love with my way
http://math.stackexchange.com/questions/1219016/indefinite-integral-with-sin-and-cos/1219078#1219078

Answer 27

other*

Answer 28

and no I didn't steal randomgirl 's thoughts :p

Answer 29

that is my fav trig identity!

Answer 30

mine too!! it is super cute.

Answer 31

wolf's solution isn't bad, it looks pretty long tho