Homework Statement
∫ 1/x arctan (lnx) dx
Homework Equations
The Attempt at a Solution
1.U substitution. SO u = ln x, du= 1/x dx
∫ arctan u du
2.by parts: u = arctan u du = 1/ 1+u^2
v = 1 dv = du
3. uv - ∫vdu = artan u - ∫ 1/ 1+u^2
=...
Homework Statement
I want to check the answer to this question.
⌠ 2 x dx / (x+1) (x+2)
⌡ 1
Homework Equations
The Attempt at a Solution
For partial fraction I got A= -1 and B = 2
My final answer is -ln 2 + ln4 - ln3 = ln 4/ ln2 * ln 3
Homework Statement
How do you prove ln(e^x) = x?
Homework Equations
The Attempt at a Solution
I just memorized this but want to know how I can prove this
Homework Statement
Find lim sin4x/3x
x -> 0
Homework Equations
The Attempt at a Solution
I did some algebraic massaging and got
sin4/3x = 4/4 * sin4x/3x = 4/3 * sin4x/4x
but I don't understand how sin4x/4x becomes 1 like my textbook says so.
Homework Statement
I just want to make sure I include all the steps in doing this:
lim (6x-7) = 11
x->3
Homework Equations
The Attempt at a Solution
given ε>0, we need to find a δ>0, such that 0< lx-3l < δ then 0 < l (6x-7)-11 l < ε
To prove this I need to make 0 < l...
Homework Statement
I need to find whether the sequence converges or diverges
1.(1+ 1/n)^2n
2.arctan x
Homework Equations
The Attempt at a Solution
1. My answer is wrong but I don't know how I can get the right answer
My answer
(1 + 1/n) ^2n becomes 1 because 1/n tends to...
I'm weak on trig. For 1 + sinx, does it shift the graph upward by 1 because y intercept is 1
when x=0? And the graph of sine goes up and down. That's why I doubt that the function increases all the time. I know that I'm wrong but I'm still wondering why.
Homework Statement
I've got two questions
1.Find the intervals on which f increases and and the intervals on which f decreases.
f(x)= x-cosx, 0≤x≤2∏
2. Why is f(x) = -2sin2x - 2sinx = -2sinx (2cosx + 1)
How can you factor 2cosx from the function on the left side?
Homework Equations...
Homework Statement
Evaluate
∫(1)to(-1) l sin(x) l dx
Homework Equations
The Attempt at a Solution
If I just leave it like this, is it wrong?
1. ∫(1)to(-1) l sin(x) l dx = l -cos (x)l
2. l -cos (1) l — l -cos (-1)l = 0 - 0 = 0
Homework Statement
I just learned this idea from my lecture in calculus. I think I understand it at a surface level but don't know much about why it is an astounding discovery as my lecturer suggested.
Homework Equations
So, it states that let (f) be a continuous on an interval [a,b]. Let...
Homework Statement
f(x) = x(x-5)^4
Homework Equations
The Attempt at a Solution
I used product rule and chain rule so,
1(x-5)^4 + 4x(x-5)^3(1)
The answer to this question is 5(x-5)^3(x-1). I left it like the answer above. I don't know how to expand (x-5)^4. Thanks.