Integration Q (1 Viewer)

[shaon0]

Can someone please help me with:
1) S (x^2+2)/(x^2+1) dx
2) S (x+1)/ (sqrt(4-x^2)) dx

 

[tommykins]

1) (x^2+2)/(x^2+1) = (x^2 + 1 + 1)/(x^2 + 1) = 1 + 1/x^2+1
Integral of that is x + atan x + c
2) Split it into x/sqrt[4-x²] + 1/sqrt[4-x²]
use u = 4-x² substitution, and the 1/sqrt[4-x²] becomes asin(x/2)

 

[shaon0]

1) (x^2+2)/(x^2+1) = (x^2 + 1 + 1)/(x^2 + 1) = 1 + 1/x^2+1
Integral of that is x + atan x + c
2) Split it into x/sqrt[4-x²] + 1/sqrt[4-x²]
use u = 4-x² substitution, and the 1/sqrt[4-x²] becomes asin(x/2)

for 2) could i substitute in x=2sinx?

 

[tommykins]

After splitting it or ?

 

[shaon0]

After splitting it or ?

before splitting it?

 

[tommykins]

Mmm, I haven't done the working but by inspection you have that +1 stuck in there and since the bottom becomes 2cosx, you have to integrate secx which I don't think you'd be able to do with your current knowledge.

 

[shaon0]

Mmm, I haven't done the working but by inspection you have that +1 stuck in there and since the bottom becomes 2cosx, you have to integrate secx which I don't think you'd be able to do with your current knowledge.

ok thanks for your help

 

[tommykins]

post if you want me to post up the solution.

 

[shaon0]

post if you want me to post up the solution.

Could you do the solution for part 2)? Also can i use partial fractions or will it be too time consuming?

 

[tommykins]

回复: Re: Integration Q

I'd say it's too time consuming.

Split it into x/sqrt[4-x²] + 1/sqrt[4-x²]
use u = 4-x² substitution, and the 1/sqrt[4-x²] becomes asin(x/2)

We'll deal with the x/sqrt[4-x²] as we've done the second part.

let u = 4-x², du = -2xdx, dx = -du/2x

int. -1/2 u^-1/2 = -1/2 u^1/2 = -1/2 sqrt[4-x^2]

Final answer is -

-sqrt[4-x^2]/2 + asin(x/2) + c

 

[shaon0]

Re: 回复: Re: Integration Q

I'd say it's too time consuming.

Split it into x/sqrt[4-x²] + 1/sqrt[4-x²]
use u = 4-x² substitution, and the 1/sqrt[4-x²] becomes asin(x/2)

We'll deal with the x/sqrt[4-x²] as we've done the second part.

let u = 4-x², du = -2xdx, dx = -du/2x

int. -1/2 u^-1/2 = -1/2 u^1/2 = -1/2 sqrt[4-x^2]

Final answer is -

-sqrt[4-x^2]/2 + asin(x/2) + c

As you can see, i really suck at maths. anyways, thanks for the solution, i would have never thought of that

 

[shaon0]

Re: 回复: Re: Integration Q

I'd say it's too time consuming.

Split it into x/sqrt[4-x²] + 1/sqrt[4-x²]
use u = 4-x² substitution, and the 1/sqrt[4-x²] becomes asin(x/2)

We'll deal with the x/sqrt[4-x²] as we've done the second part.

let u = 4-x², du = -2xdx, dx = -du/2x

int. -1/2 u^-1/2 = -1/2 u^1/2 = -1/2 sqrt[4-x^2]

Final answer is -

-sqrt[4-x^2]/2 + asin(x/2) + c

are you using the rule: S dx/x =ln(x) + c?

 

[tommykins]

回复: Re: 回复: Re: Integration Q

How stupid of me, I made a mistake.
-1/2 int. u^-1/2 = -1/2[2u^(1/2)] = -u^1/2 = -sqrt[4-x^2]

There is no /2, sorry.

 

[shaon0]

Re: 回复: Re: 回复: Re: Integration Q

How stupid of me, I made a mistake.
-1/2 int. u^-1/2 = -1/2[2u^(1/2)] = -u^1/2 = -sqrt[4-x^2]

There is no /2, sorry.

thought so. Thanks for your solution. Do solving these types of questions take practice or is it just natural?

 

[tommykins]

回复: Re: 回复: Re: 回复: Re: Integration Q

after doing alot of them, you just do it by inspection.

integrations fairly easy, just identifying a way to approach it then everything else is just simple algera.

 

[shaon0]

Re: 回复: Re: 回复: Re: 回复: Re: Integration Q

after doing alot of them, you just do it by inspection.

integrations fairly easy, just identifying a way to approach it then everything else is just simple algera.

yeah....so i should just do more questions.
btw, wats the best textbook to use for learning 4unit?

 

[shaon0]

Re: 回复: Re: 回复: Re: 回复: Re: Integration Q

i use tezza lee's text book
i think thats sufficient to learn 4 unit, then just do past hsc questions

WOOO NO MORE SCHOOL!!! =]

Ok, currently i'm using Fitzpatrick 4unit maths which is ok.
Good luck for your HSC exams

 

[shaon0]

Re: 回复: Re: 回复: Re: 回复: Re: Integration Q

the cambridge textbook is good for harder questions, i used both fitzpatrick and cambridge and thought they made a pretty good combo

what books are recommended for 4unit? Cambridge i think i'll get when i actually get taught 4unit but when i teach myself i'll use fitzpatrick because Cambridge is too hard to understand lol.

 

[conics2008]

Re: 回复: Re: 回复: Re: 回复: Re: Integration Q

shaono start using fitzpatrick first to learn the concepts etc, and then challenge your self with cambridge =]

 

[shaon0]

Re: 回复: Re: 回复: Re: 回复: Re: Integration Q

shaono start using fitzpatrick first to learn the concepts etc, and then challenge your self with cambridge =]

Yeah thats wat i plan to do.