91Ó°ÊÓ

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

Use integration by parts to find each integral. $$ \int \frac{\ln (x+1)}{\sqrt{x+1}} d x $$

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

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 solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=5 y(100-y)\) \(y(0)=10\)

Use integration by parts to find each integral. $$ \int \frac{x}{\sqrt{x+1}} d x $$

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

Find each integral by using the integral table on the inside back cover. $$ \int \sqrt{9 x^{2}+16} d x $$

Evaluate each improper integral or state that it is divergent. \(\int_{5}^{x} \frac{1}{(x-4)^{3}} d x\)

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. \(y^{\prime}=\sqrt{x+y}\)