Evaluate the following: i) int (3e^x)/(e^x -1)dx ii) int_0^1 Xcos(pi X)dX iii) lim_(xrightarrow 1)(x-1)/(sqrt(x)-1) iv) lim_(xrightarrow infty) x/(ln (1+3e^x))

Antiderivatives in complex analysis
Let U be an open set in $\mathbb{C}$ and $f:U\to E$ be holomorphic on U. Let now $\gamma$ be a path in U that begins with $z_1\in U$ and ends with $z_2\in U.$. I wonder for which assumptions the following formula ${\int }_{\gamma }\frac{df}{dz}(z)=f(z_2)-f(z_1)$ is true?
I have found many counterexamples (such as $f^{\prime }(z)=\frac{1}{z}$) where the above formula fails if the set U is not simply connected. Here is a related result.
Another question, if f is well defined on $\overline{U}$ and $z_1\in \mathrm{\partial }U$ is the above formula also true?

Solve this question
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{(x+3)}^{1+\frac{1}{x}}-x^{1+\frac{1}{x+3}}}$

Use L'Hospital Rule to evaluate the following limits.
$\underset{x\to 0}{lim}(\tan hx{)}^x$

What can be said about the sum and difference of each pair of functions?
(a) Two even functions
(b) Two odd functions
(c) An odd function and an even function

Find positive a, b for that function has limit
My problem is to find $a\ge 0$, $b\ge 0$ for for which the following limit:
$\underset{x\to +\mathrm{\infty }}{lim}(5^{\frac{1}{x}}+\sin (a{x}^3+\frac{3\pi }{2})+\ln (1+\frac{e^{bx}-e^{-bx}}{2}))$
exists and equals 0.
I think that answer is a=0, b=0. But I do not know how to accurately prove that
$\underset{x\to +\mathrm{\infty }}{lim}\cos (a{x}^3)$ exists and equals to 1 only if a=0;
$\underset{x\to +\mathrm{\infty }}{lim}\ln (1+\sinh (bx))$ exists and equals to 0 only if b=0.
Please help me to prove these two statements.

Evaluate integral:
${\displaystyle {\int }_0^{\frac{\pi }{3}}{\ln }^2\left(\frac{\sin x}{\sin (x+\frac{\pi }{3})}\right)dx}$

How do you find the average rate of change for the function ${\displaystyle f\left(x\right)=x^2+3}$ on the indicated intervals [3,4]?