91Ó°ÊÓ

In Problems 37-48, apply integration by parts twice to evaluate each integral (see Examples 5 and 6). $$ \int \ln ^{2} z d z $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{2}^{3} \frac{x^{2}+2 x-1}{x^{2}-2 x+1} d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{y}{\sqrt{16-9 y^{4}}} d y \quad\)

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} $$

In Problems 1-54, perform the indicated integrations. \(\int \cosh 3 x d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int_{1}^{5} \frac{3 x+13}{x^{2}+4 x+3} d x $$

In Problems 1-54, perform the indicated integrations. \(\int x^{2} \sinh x^{3} d x\)

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=y(1-y), y(0)=0.5 $$

In Problems 41-48, the density of a rod is given. Find c so that the mass from 0 to \(c\) is equal to 1 . Whenever possible find an exact solution. If this is not possible, find an approximation for c. (See Examples 4 and 5 ). $$ \delta(x)=\frac{1}{x+1} $$

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=\frac{1}{10} y(12-y), y(0)=2 $$