Chapter 7

In Problems 1-14, solve each differential equation. $$ \sin x \frac{d y}{d x}+2 y \cos x=\sin 2 x ; y=2 \text { when } x=\frac{\pi}{6} $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int \arctan 5 x d x $$

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{\pi / 4} \frac{\cos x}{1+\sin ^{2} x} d x\)

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x y, y(1)=1,[1,2] $$

In Problems 1-28, perform the indicated integrations. \(\int \sin ^{4}\left(\frac{w}{2}\right) \cos ^{2}\left(\frac{w}{2}\right) d w\)

\(\int \frac{2 z-3}{\sqrt{1-z^{2}}} d z\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{6 x^{2}+22 x-23}{(2 x-1)\left(x^{2}+x-6\right)} d x $$

A tank contains 20 gallons of a solution, with 10 pounds of chemical \(A\) in the solution. At a certain instant, we begin pouring in a solution containing the same chemical in a concentration of 2 pounds per gallon. We pour at a rate of 3 gallons per minute while simultaneously draining off the resulting (well- stirred) solution at the same rate. Find the amount of chemical \(\mathrm{A}\) in the tank after 20 minutes.

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

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{3 / 4} \frac{\sin \sqrt{1-x}}{\sqrt{1-x}} d x\)

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=-2 x y, y(1)=2,[1,2] $$

In Problems 1-28, perform the indicated integrations. \(\int \sin 3 t \sin t d t\)

\(\int_{0}^{\pi} \frac{\pi x-1}{\sqrt{x^{2}+\pi^{2}}} d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{3}-6 x^{2}+11 x-6}{4 x^{3}-28 x^{2}+56 x-32} d x $$

A tank initially contains 200 gallons of brine, with 50 pounds of salt in solution. Brine containing 2 pounds of salt per gallon is entering the tank at the rate of 4 gallons per minute and is flowing out at the same rate. If the mixture in the tank is kept uniform by constant stirring, find the amount of salt in the tank at the end of 40 minutes.

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

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

Apply Euler's Method to the equation \(y^{\prime}=y, y(0)=1\) with an arbitrary step size \(h=1 / N\) where \(N\) is a positive integer. (a) Derive the relationship \(y_{n}=(1+h)^{n}\). (b) Explain why \(y_{N}\) is an approximation to \(e\).

In Problems 1-28, perform the indicated integrations. \(\int x \cos ^{2} x \sin x d x\) Hint: Use integration by parts.

In Problems 17-26, use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. 17\. \(\int \frac{d x}{\sqrt{x^{2}+2 x+5}}\)

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

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. $$ \int_{1}^{e} \sqrt{t} \ln t d t $$

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 \(y_{0}=0\).

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

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

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

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

In Problems 1-54, perform the indicated integrations. \(\int \frac{\sin \left(\ln 4 x^{2}\right)}{x} d x\)

In Problems 1-28, perform the indicated integrations. \(\int \tan ^{4} x d x\)

\(\int \frac{3 x}{\sqrt{x^{2}+2 x+5}} d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{4}+8 x^{2}+8}{x^{3}-4 x} d x $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int z^{3} \ln z d z $$

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

In Problems 1-28, perform the indicated integrations. \(\int \cot ^{6} x d x\)

\(\int \frac{2 x-1}{\sqrt{x^{2}+4 x+5}} d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{6}+4 x^{3}+4}{x^{3}-4 x^{2}} d x $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int t \arctan t d t $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{6 e^{x}}{\sqrt{1-e^{2 x}}} d x\)

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

$$ \int \sqrt{5-4 x-x^{2}} d x $$

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

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int \arctan (1 / t) d t $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{x}{x^{4}+4} d x\)

In Problems 1-28, perform the indicated integrations. \(\int \cot ^{3} 2 t d t\)

\(\int \frac{d x}{\sqrt{16+6 x-x^{2}}}\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{5 x+7}{x^{2}+4 x+4} d x $$

Suppose that tank 1 initially contains 100 gallons of solution, with 50 pounds of dissolved salt, and tank 2 contains 200 gallons, with 150 pounds of dissolved salt. Pure water flows into tank 1 at 2 gallons per minute, the well-mixed solution flows out and into tank 2 at the same rate, and finally, the solution in tank 2 drains away also at the same rate. Let \(x(t)\) and \(y(t)\) denote the amounts of salt in tanks 1 and 2, respectively, at time \(t\). Find \(y(t)\). Hint: First find \(x(t)\) and use it in setting up the differential equation for tank \(2 .\)