# Integral of t/1+t^2

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..

## Answers and Replies

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.

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?

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)

Last edited:
rojak44
could anyone help me here? pls..

Homework Helper
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?

Homework Helper
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?

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

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

Homework Helper
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.

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