## angela210793 3 years ago @satellite73

1. angela210793

|dw:1341334653707:dw|

2. satellite73

ick i think the idea is to write the denominator as a single trig function

3. angela210793

:O i thought maybe to split and take 2 fractions O.o

4. angela210793

but idk how to do tht -_-

5. satellite73

it is going to be $$5\sin(x+\theta)$$ but i don't see a nice form for $$\theta=\tan^{-1}(\frac{4}{3})$$

6. satellite73

actually it doesn't matter because $$\theta$$ is a constant also i made a mistake, it is $$5\sin(x-\theta)$$ still no matter

7. satellite73

i don't think there is a snappier way to do this. let me think for a second

8. angela210793

okok

9. satellite73

ok no i can't think of a better way, and also i tried wolfram and what a disaster what you need to know is $a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\theta)$ where $$tan(\theta)=\frac{b}{a}$$

10. angela210793

hmmmm....suppose i know tht....how to use it O.o

11. satellite73

well then it is easy, especially if you have a table of integrals this becomes $\frac{1}{5}\int\csc\left(x-\tan^{-1}(\frac{4}{3})\right)$

12. satellite73

don't worry about the arctan part, that is a number

13. satellite73

so all you need to do is look up the anti derivative of cosecant and you are done

14. satellite73

here is a video explanation http://www.youtube.com/watch?v=STUh3ni4l50

15. angela210793

hmmmm....we've never use secant...wht is it?

16. satellite73

cosecant it is the reciprocal of sine

17. satellite73

integral is $\int csc(x)dx=-\ln(\cot(x)+\csc(x))$ if you have not used this i really have no idea how you are supposed to do this problem maybe multiply top and bottom by the conjugate?

18. angela210793

:/....ok.. thank you Sir :D

19. satellite73

maybe we can get some help i will repost there might be a snappy trick

20. angela210793

okk

21. satellite73

maybe myininaya has a snappy way multiply top and bottom by $$3\sin(x)+4\cos(x)$$ maybe?

22. myininaya

I don't know @satellite73 I like what you did.

23. satellite73

ok then i will stick to that. thanks

24. myininaya

I actually think what sat did was probably the most snappiest thing you can do for this type of integral

25. waterineyes

If we replace sinx by 2tan(x/2)/(1 + tan^2(x/2),

26. angela210793

sec=1/cosx????

27. myininaya

yes @angela210793 sec(x)=1/(cos(x)) csc(x)=1/(sin(x))

28. angela210793

ok....thanks guys :D

29. waterineyes

$\huge sinx = \frac{2\tan \frac{x}{2}}{1+ \tan^2\frac{x}{2}}$ $\huge cosx = \frac{1 - \tan^2\frac{x}{2}}{1+ \tan^2\frac{x}{2}}$