91Ó°ÊÓ

Use integration by parts to find each integral. $$ \int \frac{\ln t}{\sqrt{t}} d t $$

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

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. \(y^{\prime}=x^{m} y^{n} \quad\) (for \(m>0, n \neq 1\) )

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

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

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

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

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

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