Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2010 / 5 / ii / a

An entire antiderivative is $F(z)=z\sin z+\cos z$, since differentiation cancels the sine terms and leaves $F^{\prime }(z)=z\cos z$. The curve runs from $-i$ to $i$. Both terms of $F$ are even, so $F(i)=F(-i)$, and the complex line integral gives