# Integral of t/1+t^2

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.. 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
Homework Helper
$$\frac{t}{1+t^2} dt$$

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

here it is,pls correct me if im 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)

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could anyone help me here? pls..

Defennder
Homework Helper
here it is,pls correct me if im 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.

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
am i correct? btw if we remove t,it will become 1/2u right since it was 1t/2tu?

Defennder
Homework Helper
Yes, that is correct.

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
Homework Helper
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 $$\frac{d}{dt} \ e^{mt} = me^{mt}$$.

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

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

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

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