91Ó°ÊÓ

$$ \text { Use integration by parts to find each integral. } $$ $$ \int t e^{-0.5 t} d t $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int t e^{-0.2 t} d t $$

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

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. $$ \int_{1}^{2} x^{3} d x $$

Find each integral by using the integral table on the inside back cover. $$ \int x^{99} \ln x d x $$

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. $$ y^{\prime}=x y^{2} $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=48(2-y)\) \(y(0)=0\)

Find each integral by using the integral table on the inside back cover. $$ \int x^{-101} \ln x d x $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=0.05(0.25-y)\) \(y(0)=0\)

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. $$ y^{\prime}=x^{2} y $$