Solving an Anti-Derivative Problem with Trigonometric Identities

[zhen]

find the anti-derivative of :

f(x) = 4 - 3(1+x^2)^(-1)

I have thought this question for hours...but no clue at all...

that is what I have attempted:

F(x) = 4x - 3 Ln(1 + x^2) ...
but if i differentiated it ---
then I got F'(x) = 4 - 3*(2x)/(1 + x^2)...

is there anyway to eliminate the (2x) ...?

please help

 

[HallsofIvy]

Well, first the "4" is obvious- you are completely correct that it's anti-derivative is 4x.

Now, for 3/(1+ x^2). Log doesn't work because 1+ x^2 is not x!

Do you know the derivative of arctan(x)?

 

[zhen]

oh...thank you...
I totally forgot about there are some formula for that...
yes...
so is the answer 4x - 3 arc tan x + C?

 

[zhen]

but in my textbook, i can not find the prove of those identities...
just wonder if there is any link for that...