91Ó°ÊÓ

Find each integral by using the integral table on the inside back cover. $$ \int x^{3} e^{2 x} d x $$

Evaluate each improper integral or state that it is divergent. \(\int_{2}^{\infty} 3 x^{-4} d x\)

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. \(y^{\prime}=\frac{x}{x^{2}+1}\)

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=2(100-y)\) \(y(0)=0\)

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

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. \(\int_{0}^{\infty} e^{-t} d t\)

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