Find an upper bound for $|\int_C(\cos z+\frac{e^z}{z+1})dz|$, where C is the circle $|z+i|=4$
Find a lower bound for $|\int_C(\cos z+\frac{e^z}{z+1})dz|$, where C is the circle $|z|=4$
The integral of $\cos z$ is $\sin z$ and the integral of $\frac{e^z}{z+1}$ with respect to $z$ is (according to Wolphram Alpha) $Ei(z+1)/e + C$ where $Ei(x)$ is the exponential integral Ei.
What is the upper bound OR lower bound ?
Obviously I couldn't get any further that's why I am asking the experts...
Don't use primitives.
You have to apply this: If $\gamma$ is a circle and $r$ is its radius, $f(z)$ is a holomorphic function in an open set that contains the circle, and $|f(z)|\leq M$ everywhere in the circle, then,
$$\left|\int_\gamma f(z)dz\right|\leq2\pi rM$$