Solve Integral of t/1+t^2 - 1/2 ln (1+t^2) + c

[rojak44]

hello there,im having trouble with this question..
find int t/1+t^2 dt

my answer is 1/2 ln t (1+t^2)+c
but the answer that i have copy is 1/2 ln (1+t^2)+c
maybe i have copy it wrongly,can someone tell me which one is right?
thx..

 

[exk]

How did you get your answer?

 

[rojak44]

im using the integration using substitution..
im confuse right now.. can anyone give me the full calculation or what method i should use.

 

[rock.freak667]

$$\frac{t}{1+t^2} dt$$

Let $u=1+t^2$ what is $\frac{du}{dt}$? then what is dt in terms of du?

 

[rojak44]

here it is,pls correct me if I am wrong.
int t/1+t^2 dt
u = 1+t^2
du/dt = 2t
dt = 1/2t du
= int (t/u)(1/2t du)
= int t/2tu du
= 1/2 int t/tu du
= 1/2 int t(tu)^-1 du
= 1/2 int t^0 u^-1 du
= 1/2 [t^1 ln u]+c
= 1/2 ln t (1+t^2)+c (answer)

 

[rojak44]

could anyone help me here? pls..

 

[Defennder]

here it is,pls correct me if I am wrong.
int t/1+t^2 dt
u = 1+t^2
du/dt = 2t
dt = 1/2t du
= int (t/u)(1/2t du)
= int t/2tu du
= 1/2 int t/tu du
= 1/2 int t(tu)^-1 du
= 1/2 int t^0 u^-1 du
= 1/2 [t^1 ln u]+c
= 1/2 ln t (1+t^2)+c (answer)

Cancel out t in this step.

 

[rojak44]

u mean remove the t? so it will become 1/2u?
here,
= int 1/2u du
= int u^-1/2 du
= 1/2 int u^-1 du
=1/2 [ln u]+c
= 1/2 ln (1+t^2)+c (answer)
am i correct? btw if we remove t,it will become 1/2u right since it was 1t/2tu?

 

[Defennder]

Yes, that is correct.

 

[rojak44]

thank you! :)
i got one more question,
differentiate y= 2m e^mt/cos 2m
is it dy/dt or dy/dm?
2m e^mt,
e^mt if differentiate = e^mt or me^mt?

 

[Defennder]

I can't read what you are writing here, is it supposed to be $$y = \frac{2me^{mt}}{cos(2m)}$$?

Is m a constant or function of t?

As for your 2nd question, it is true that $$\frac{d}{dt} \ e^{mt} = me^{mt}$$.

 

[rojak44]

it is correct except the cos have no bracket, cos2m
i have no idea,im having trouble to decide whether it is dy/dm or dy/dt

 

[Defennder]

Is m a function of t? Secondly what are you differentiating with respect to?

 

[rojak44]

here is my answer,do tell whether it is right or wrong.
2e^mt(m cos 2m+cos 2m+2m sin 2m)/(cos 2m)^2

 

[rojak44]

just assume it as dy/dt
sorry I am not good in english.
im using quotient rule to solve that.

 

[Defennder]

You haven't clarified this question: Is 'm' a constant? Or is it a function of t?