91Ó°ÊÓ

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

A tank initially contains 120 gallons of pure water. Brine with 1 pound of salt per gallon flows into the tank at 4 gallons per minute, and the well- stirred solution runs out at 6 gallons per minute. How much salt is in the tank after \(t\) minutes, \(0 \leq t \leq 60\) ?

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int_{1}^{e} \sqrt{t} \ln t d t $$

A tank initially contains 50 gallons of brine, with 30 pounds of salt in solution. Water runs into the tank at 3 gallons per minute and the well- stirred solution runs out at 2 gallons per minute. How long will it be until there are 25 pounds of salt in the tank?

In Problems 1-28, perform the indicated integrations. \(\int x \sin ^{3} x \cos x d x\)

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int_{1}^{5} \sqrt{2 x} \ln x^{3} d x $$

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{3}+x^{2}}{x^{2}+5 x+6} d x $$

\(\int \frac{d x}{\sqrt{x^{2}+4 x+5}}\)

In Problems 1-54, perform the indicated integrations. \(\int \frac{x^{3}+7 x}{x-1} d x\)

Suppose that the function \(f(x, y)\) depends only on \(x\). The differential equation \(y^{\prime}=f(x, y)\) can then be written as $$ y^{\prime}=f(x), \quad y\left(x_{0}\right)=0 $$ Explain how to apply Euler's Method to this differential equation if \(y_{0}=0\).