How do solve $\int\frac{2\sin(x)+3\cos(x)}{3\sin(x)+2\cos(x)}dx$?

Question

How do I solve this integral? Should I use some kind of an integral substitution?

Are you sure the denominator has no negative sign for the sin x. If that is the case see my hint below otherwise just substitute the numerator as the denominator is its derivative. — , Nov 20 '16 at 17:02
http://math.stackexchange.com/questions/1219016/indefinite-integral-with-sin-and-cos-int-frac3-sinx-2-cosx2-sin/1219034#1219034 — tired, Nov 20 '16 at 17:05
http://math.stackexchange.com/questions/2010770/finding-the-integral-int-0-large-frac-pi4-frac-cosx-dxa-cosx — Victor Chen, Nov 20 '16 at 17:17

score 5 · Accepted Answer · answered Nov 20 '16 at 17:03

5

Hint. By setting $$ I=\int\frac{\cos(x)\:dx}{2\cos(x)+3 \sin(x)}\quad J=\int\frac{\sin(x)\:dx}{2\cos(x)+3 \sin(x)} $$ One may observe that

$$\begin{cases} 2 I+3J=\displaystyle\int 1\:dx \\ 3 I-2J=\displaystyle \int\frac{(2\cos(x)+3 \sin(x))'}{2\cos(x)+3 \sin(x)}\:dx \end{cases} $$ Can you take it from here?

answered Nov 20 '16 at 17:03

Olivier Oloa

122,789

3

You should give credit to the person who had this idea. – Git Gud Nov 20 '16 at 17:04
1

@Git Gud Please let me know the name. – Olivier Oloa Nov 20 '16 at 17:06
1

As if you didn't know Tao did it first... – Git Gud Nov 20 '16 at 17:09
@Git Gud Please, I don't see any "Tao" in your link. – Olivier Oloa Nov 20 '16 at 17:11
1

This is actually quite an old chestnut. See question 3 in http://pmt.physicsandmathstutor.com/download/Maths/STEP/Advanced%20Problems%20in%20Mathematics%20(STEP).pdf. But it probably dates from before then. – David Quinn Nov 20 '16 at 17:16
1

I linked to a very specific comment by Wong which reads: "In this math education article the author describes giving the same problem to a young Terence Tao, aged 8; he gave essentially the same beautiful solution". And I don't believe for one minute that you didn't already know the trick. – Git Gud Nov 20 '16 at 17:16
1

@Git Gud I did not know Tao evaluated it at the age of 8... If this is true, that's impressive. Thanks for the information. As a side note, I don't think I'm the only one not being aware of this. – Olivier Oloa Nov 20 '16 at 17:21
1

@GitGud This was never a result that was important enough for a publication. Do you cite Babylonians every time you apply an operation on both sides of an equation? Didn't think so. – John11 Nov 20 '16 at 17:25
1

@John11 You miss the point. When a 70K+ user posts a well known trick as an answer to a calculus question in order to reap the fruit, you can count on some people here to try to discourage this. I'm one of them. – Git Gud Nov 20 '16 at 17:28
1

@GitGud This has degenerated into a longer than necessary sequence of comments that could have been avoided by just stating directly the first time, where you know this trick from, to benefit the reader. – John11 Nov 20 '16 at 17:36

score 0 · Answer 2 · answered Nov 20 '16 at 17:08

0

If there is no minus sign, multiply the numerator and the denominator by $ sec (x)^{3} $. Then substitute $ u= tan (x) $ and then the integral becomes easy. Hope it helps.

answered Nov 20 '16 at 17:08

score 0 · Answer 3 · answered Nov 20 '16 at 17:12

0

Hint:

Let $$2\sin x+3\cos x=A(3\sin x+2\cos x)+B\cdot\dfrac{d(3\sin x+2\cos x)}{dx}$$

answered Nov 20 '16 at 17:12

lab bhattacharjee

279,016

How do solve $\int\frac{2\sin(x)+3\cos(x)}{3\sin(x)+2\cos(x)}dx$?

3 Answers3