91Ó°ÊÓ

Evaluate each improper integral or state that it is divergent. $$ \int_{1}^{\infty} \frac{1}{x^{1.01}} d x $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int s(2 s+1)^{4} d s $$

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. $$ \int_{0}^{1} \sqrt{1+x^{2}} d x $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=80-2 y\) \(y(0)=0\)

$$ \text { Use integration by parts to find each integral. } $$ $$ \int \frac{x+1}{e^{3 x}} d x $$

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. $$ \int_{0}^{1} \sqrt{1+x^{3}} d x $$

Evaluate each improper integral or state that it is divergent. $$ \int_{10}^{\infty} e^{-x / 5} d x $$

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{x(x-3)} d x $$

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. $$ y^{\prime}=x^{m} y \quad \text { (for } m>0 \text { ) } $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=27-3 y\) \(y(0)=0\)