91Ó°ÊÓ

Evaluate each improper integral or state that it is divergent. $$ \int_{0}^{\infty} e^{-0.05 t} d t $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int \frac{x}{e^{2 x}} d x $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=2 \sqrt{y} $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=2-0.01 y\) \(y(0)=0\)

Estimate each definite integral "by hand," using Simpson's Rule with \(n=4\). Round all calculations to three decimal places. Exercises \(19-26\) correspond to Exercises \(1-8\), in which the same integrals were estimated using trapezoids. If you did the corresponding exercise, compare your Simpson's Rule answer with your trapezoidal answer. $$ \int_{0}^{1} e^{-x^{2}} d x $$

Find each integral by using the integral table on the inside back cover. $$ \int \frac{z}{z^{4}-4} d z $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=6-8 y\) \(y(0)=0\)

Estimate each definite integral "by hand," using Simpson's Rule with \(n=4\). Round all calculations to three decimal places. Exercises \(19-26\) correspond to Exercises \(1-8\), in which the same integrals were estimated using trapezoids. If you did the corresponding exercise, compare your Simpson's Rule answer with your trapezoidal answer. $$ \int_{0}^{1} e^{x^{2}} d x $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=5+y $$

Find each integral by using the integral table on the inside back cover. $$ \int \frac{z}{9-z^{4}} d z $$