## abz_tech 3 years ago integration question

1. abz_tech

$\int\limits_{3/-2}^{1/-2} dx/4x^{2}+12x+13$

2. TuringTest

complete the square in the denominator and then trig substitute

3. abz_tech

i tried that but.. got stuck at the +13

4. abz_tech

got $(x+2)^{2}+8$

5. abz_tech

8 is not a square and if u factorise (x+2)^2 u dont get 4x^2....

6. TuringTest

factor that four out of the denom first$\int_{-\frac32}^{-\frac12}\frac{dx}{4x^2+12x+13}=\frac14\int_{-\frac32}^{-\frac12}\frac{dx}{x^2+3x+\frac{13}4}$$=\frac14\int_{-\frac32}^{-\frac12}\frac{dx}{x^2+3x+\frac94+1}$

7. TuringTest

$=\frac14\int_{-\frac32}^{-\frac12}{dx\over(x+\frac32)^2+1}$

8. abz_tech

how did u get 9/4+1? =/

9. TuringTest

13/4=(9/4)+1 completing the square on x^2+3x means adding a term of 9/4 you can save some trouble by noticing that 13/4=1+9/4 so we can make our constant term from the fraction we already have

10. abz_tech

can u continue plz

11. TuringTest

trig sub:$x+\frac32=\tan\theta\implies dx=\sec^2\theta d\theta$change bounds...

12. abz_tech

=/

13. dpaInc

|dw:1338250956092:dw|

14. abz_tech

to be honest i just want to know how my teacher got dx/4x^2+12x+13 to be dx/(2x+3)^2+2^2

15. abz_tech

$4x ^{2}+12x+13 \to (2x+3)^{2}+2^2$ can u tell me how he got it =/..thx

16. TuringTest

complete the square$4x^2+12x+13=4x^2+12x+9+4=(2x+3)^2+2^2$knowing that splitting the terms up like this is not as obvious as the way in which I did it I think, so it requires a little more intuition I prefer to get the coefficient of the x term to be 1 to be safe, but notice that you can take my answer and your teachers result follows if you let the 4 back in the denom

17. TuringTest

he just noticed that splitting the 13=9+4 would work

18. TuringTest

above I derived it my way$=\frac14\int_{-\frac32}^{-\frac12}\frac{dx}{x^2+3x+\frac94+1}$let back in the 4 now...$=\int_{-\frac32}^{-\frac12}\frac{dx}{4x^2+12x+9+4}$from whence follows your teacher's way note both will give the same answer obviously

19. abz_tech

alright thx.. but the way u did is not in my course =/ i dont even know what changing bounds means..yet lol

20. TuringTest

the bounds in terms of x are -3/2 and -1/2 we make the substitution $2x+3=2\tan\theta$plug in the bounds of x and we can get them in terms of theta$2(-\frac32)+3=0=2\tan\theta\implies \theta=0$$2(-\frac12)+3=2=2\tan\theta\implies\theta=\tan^{-1}1=\frac{\sqrt2}2$so the new bounds are$\int_{0}^{\frac{\sqrt2}2}\text{(whatever)}dx$

21. abz_tech

:o..never did that.. before :P..kinda confusing.. we jsut call what u call bounds..limits

22. TuringTest

whatever floats your boat I just don't like getting that term confused with limits like$\lim_{x\to a}f(x)$

23. TuringTest

call 'em Freddies for all I care though, makes no mathematical difference whatever is easiest for you to keep straight

24. abz_tech

lol.. ya i guess, thx again see u around

25. TuringTest

welcome, good luck :)