## anonymous one year ago need help in the derivation of integration formula:

1. misty1212

HI!!

2. misty1212

i didn't know there was such a formula

3. anonymous

$\int\limits_{.}^{.} \sqrt{x^{2}-a^{2}}dx =\frac{x }{ 2 }\sqrt{x ^{2}-a ^{2}}-\frac{ a ^{2} }{ 2 }\log \left| x+\sqrt{x ^{2}+a^2} \right|+C$

4. anonymous

solve by taking x=a sect

5. anonymous

no problem @misty1212

6. misty1212

this one is a pain in the neck i think, but the sub is right

7. misty1212

you are going to get $a^2\int \sec^3(u)-\sec(u)du$

8. misty1212

did you get to that part, or no?

9. anonymous

yes

10. anonymous

i dont know what to do after that.....

11. misty1212

this is the kind of thing you look up in the back of the text because it is boring beyond belief

12. anonymous

can u go after that

13. misty1212

there is a "reduction formula" for secant

14. misty1212

are you allowed to use it?

15. anonymous

u can take the snapshot for the things u solved in the notebook instead of typing them. it could be easier, if u can do it.....

16. anonymous

what is reduction formula

17. misty1212

you mean in general or "what is the reduction formula for $$\int \sec^n(x)dx$$?

18. anonymous

then what is that?

19. misty1212

here are a bunch of them http://archives.math.utk.edu/visual.calculus/4/recursion.2/

20. misty1212

in your case $$n=3$$ so you get $\int sec^3(u)du=\frac{1}{2}\tan(u)\sec(u)+\frac{1}{2}\int \sec(u)du$

21. anonymous

hav u done it by taking integration by parts?

22. misty1212

it looks like that right? but it is not it comes from doing some trig business and a u sub

23. misty1212

like the same trick they use for $$\int\sin^n(x)dx$$ we can work through it if you like although it is not that interesting

24. misty1212

oh to answer your question, no i used nothing but the"reduction formula" with $$n=3$$ the one i sent the link to

25. misty1212

ignoring the annoying $$a^2$$ out front, and combining like terms, we should be at $\frac{a^2}{2}\sec(u)tan(u)-\frac{a^2}{2}\int \sec(u)du$

26. anonymous

i dont know these reduction formulas...it's not in my book....in my book, the derivation was given by the method of integration by parts and in the end it was written that u can take x=sec t to solve the problem...so i did it by doing the substitution and got clutched in the middle...

27. misty1212

now how to integrate secant, again it is not interesting, best to memorize it however, you want a good explanation, easier than i can write here, click on this http://math2.org/math/integrals/tableof.htm then go to "proof"

28. anonymous

i can prove by the method of integration by parts...so no problem, when i would learn about these reduction formulas then i would go by that method also...

29. misty1212

i think you use parts for $$\int \sec^3(x)dx$$ but not for $$\int \sec(x)dx$$

30. misty1212

it is just something they do to make a formula is all some people like formulas

31. ganeshie8

if you like partial fractions, $\sec x =\dfrac{\cos x}{\cos^2x} = \dfrac{\cos x}{1-\sin^2x}$

32. misty1212

oh cool, lots easier than the un-intuitive multiplying top and bottom business

33. anonymous

please give a more detail where to start from....@ganeshie8

34. ganeshie8

same can be extended to $$\sec^3x$$ too i think $\sec^3 x =\dfrac{\cos x}{\cos^4x} = \dfrac{\cos x}{(1-\sin^2x)^2}$

35. misty1212

@ganeshie8 is proving $\int\sec(x)dx=\ln(\sec(x)+\tan(x))$

36. anonymous

i dont know how to integrate that partial fraction....@ganeshie8

37. misty1212

$\int \frac{\cos(x)}{1-\sin^2(x)}dx$ put $$u=\sin(x)$$ and integrate $-\int \frac{du}{1-u^2}$ using partial fractions

38. misty1212

@ganeshie8 that is correct yes? never saw it done this way

39. anonymous

oh yes i got that...

40. ganeshie8

looks good to me !

41. ganeshie8

for $$\int\sec^3x\,dx$$ i would try reduction formula though as partial fractions looks a bit lengty http://www.wolframalpha.com/input/?i=%5Cint+1%2F%281-u%5E2%29%5E2

42. anonymous

the problem is that i dont know these reduction formulas and the link that @ganeshie8 has given....i am a beginner in integration i only know the formulas given in my book....i wold rather use integration by parts from the beginning than using all such formulas...that is easiest...

43. anonymous

the statement in the middle had initiated me...@ganeshie8

44. anonymous

@ganeshie8