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How to integrate $\displaystyle\int\frac{\sin ^{2}\theta }{\cos ^{5}\theta }d\theta$? This is also homework,how to start with it?

I try to change into

$$\int\tan ^{2}\theta \sec ^{3}\theta d\theta$$

and then

$$\int\tan ^{}\theta \sec ^{}\theta \; \tan \theta \sec ^{2}\theta d\theta$$

if I set $u=\sec \theta $ then I will deal with a single $ \tan \theta $ then I stuck

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You just asked and got an answer to a very similar question. Why not show us what happens when you try to apply the method of that other question to this one? –  Gerry Myerson Jan 25 '13 at 11:58
    
sorry,wait a minute –  Ave Maleficum Jan 25 '13 at 12:00
    
Good! Now we can see that some new idea is needed. But I think this one is going to turn out to be quite a bit harder. –  Gerry Myerson Jan 25 '13 at 12:14
    
actually there could be infinity problems like this one..You just change the index here.I think someone must work this out. –  Ave Maleficum Jan 25 '13 at 12:19
1  
Another trick to put in your bag is Weierstraß substitution. I doubt it's of much help here, though, since it looks like you'd end up with a very ugly rational function. –  kahen Jan 25 '13 at 12:42
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3 Answers 3

up vote 4 down vote accepted

$\int \dfrac{\sin^2\theta}{\cos^5\theta}\mathscr{d}\theta=\int \dfrac{1}{\cos^5\theta}\mathscr{d}\theta-\int \dfrac{1}{\cos^3\theta}\mathscr{d}\theta$
Let $I_n:=\int \cos^n\theta\mathscr{d}\theta$,
we have
$\begin{align*} I_n=\int \cos^{n-1}\theta\mathscr{d}\sin\theta&=\cos^{n-1}\theta\sin\theta-\int \sin\theta\mathscr{d}\cos^{n-1}\theta \\&=\cos^{n-1}\theta\sin\theta+(n-1)\int\cos^{n-2}\theta\sin^2\theta\mathscr{d}\theta \\&=\cos^{n-1}\theta\sin\theta+(n-1)(I_{n-2}-I_n) \end{align*}$
hence, $nI_n=(n-1)I_{n-2}+\cos^{n-1}\theta\sin\theta$.
Also, we have $I_{-1}=\int \sec\theta\mathscr{d}\theta=\ln |\sec\theta+\tan\theta|+C$.
So, $I_{-3}=\dfrac{\ln |\sec\theta+\tan\theta|+\sec^2\theta\sin\theta}{2}+C$ and $I_{-5}=\dfrac{3I_{-3}+\sec^4\theta\sin\theta}{4}$.
$\therefore \int \dfrac{\sin^2\theta}{\cos^5\theta}\mathscr{d}\theta=I_{-5}-I_{-3}=\dfrac{-\ln |\sec\theta+\tan\theta|-\sec^2\theta\sin\theta+2\sec^4\theta\sin\theta}{8}+C$

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Thank you Sir.Although I am sure this question is way beyond my capacity.Will this method be useful for any $\int_{}^{}{\frac{\sin ^{n}\theta }{\cos ^{m}\theta }}d\theta$ ? –  Ave Maleficum Jan 25 '13 at 12:34
    
@AveMaleficum: Let $I(n,m)=\int \dfrac{\sin^n\theta}{\cos^m\theta}\mathscr{d}\theta$, I've compute the recurrence relation in similar way: $I\(n,m)=\dfrac{\sin^{n+1} \theta}{(m-1)\cos^{m-1}\theta}-\dfrac{n-m+2}{m-1} I(n,m-2)$. You may find similar recurrence relations(with $I(n-2,m)$ or $I(n-2,m-2)$ eg.) by yourself. Try it! ;-) –  Shane Chern Jan 25 '13 at 12:53
    
Thank you,Sir.This really helps a lot. –  Ave Maleficum Jan 25 '13 at 13:06
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After taking $ u = sec \theta $ in the following:

$$ I = \int_{}^{}{\tan ^{}\theta \sec ^{}\theta \; \tan \theta \sec ^{2}\theta d\theta } $$

You'll have:

$$ I = \int_{}^{} {u^2 * \sqrt{u^2 - 1} du } = u^2 \int{\sqrt{u^2 - 1} du} - 2 \int{u * \left(\int{\sqrt{u^2 - 1} du}\right) du}$$

Edit

The following image is the assumption that $ u = \sec \theta $

triangle conversion

Please excuse my MS Paint skills.

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Sir,how did the last step work? –  Ave Maleficum Jan 25 '13 at 12:25
    
Integration by parts. But I'm not convinced that what's done here is progress towards an answer. –  Gerry Myerson Jan 25 '13 at 12:30
    
@GerryMyerson Why? –  hjpotter92 Jan 25 '13 at 12:31
    
Well, keep going and see where it gets you. For one thing, a standard way to handle integrals involving $\sqrt{u^2-1}$ is to make the trig substitution $u=\sec\theta$ --- getting you right back to where you started. –  Gerry Myerson Jan 25 '13 at 22:57
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  1. $\int\sec{x} \ dx=\ln|\sec x+\tan x| , \int \sec^2{x} \ dx=\tan x$ .
  2. $\int\dfrac{\sin^2 x}{\cos^2 x} \ dx=\int\sec^3 x+\sec^5 x \ dx$ .

If $I=\int \sec^5 x \ dx=\int \sec^3 \sec^2 x \ dx=\int \sec^3 (\tan x)' \ dx$
then $I=\sec^3 x\tan x -\int3\sec^2x \sec x\tan x\tan x \ dx =\\ \sec^3 x\tan x -\int3\sec^3x \tan^2 x=\sec^3 x\tan x -3I+3\int\sec^3x \ dx...$

Do the same for $\int \sec^3 x$ and use 1.

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The integrals of the odd powers of secant are not all that easy. I think they take more than "integration by parts". –  Gerry Myerson Jan 25 '13 at 12:32
    
@GerryMyerson: I believe now is better. Is there a difference for the integrals of even powers of secant? I believe there isn't. For any power you can calculate $\int \sec^n x \ dx$ recursively. –  P.. Jan 25 '13 at 12:49
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Even powers are much easier because they can be written as $P(\tan\theta)d(\tan\theta)$ for some polynomial $P$. –  Gerry Myerson Jan 25 '13 at 22:58
    
You are right @GerryMyerson. Thanks. –  P.. Jan 26 '13 at 6:41
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