91Ó°ÊÓ

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{2}^{4} x e^{-x / 2} d x $$

Evaluate the given integral along the indicated contour. \(\oint_{C} \operatorname{Re}(z) d z\), where \(C\) is the circle \(|z|=1\)

Show that \(\oint_{C} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\) $$ f(z)=\frac{z^{2}-9}{\cosh z} $$

Evaluate the given integral along the indicated contour. \(\oint_{C}\left(\frac{1}{(z+i)^{3}}-\frac{5}{z+i}+8\right) d z\), where \(C\) is the circle \(|z+i|=1,0 \leq t \leq 2 \pi\)

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{1}^{e} \ln x d x $$

Evaluate the given integral along the indicated contour. \(\int_{C}\left(x^{2}+i y^{3}\right) d z\), where \(C\) is the straight line from \(z=1\) to \(z=i\)

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{2}^{4} \frac{d x}{x^{2}-6 x+5} $$

In Problems 9-12, find the velocity field \(\mathbf{F}(x, y)\) of the flow of an ideal fluid determined by the given analytic function \(g(z)\). $$ g(z)=(1+i) z^{2} $$

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{2}^{4} \frac{2 x-1}{(x+3)^{2}} d x $$

In Problems 11-14, evaluate the line integrals \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\), and \(\int_{C} G(x, y) d s\) on the indicated curve \(C\). $$ G(x, y)=2 x y ; x=5 \cos t, y=5 \sin t, 0 \leq t \leq \pi / 4 $$