Q: Evaluate the indefinite integral $$\int \frac{dx}{x^2$ + 6x + 25}.$$

$$\frac{1}{4} \tan$$^{-1}$$ \left( \frac{x+3}{4} \right) + C$$

Question 1

A pool initially contains 1000 liters of water. Water flows into the pool at a rate of $$r(t) = 100 + 20 \sin(\pi t)$$ liters per minute. How much water will the pool contain after 30 minutes?

Accepted Answer

$$4000$$ liters

Question 2

Find the area between the curves $$y = \sin 2x$$ and $$y = \cos x on $$the interval $$[0, \frac{\pi}{2}].$$

Accepted Answer

$$\frac{1}{2}$$

Question 3

A solid has a base in the $$xy-$$plane bounded by $$y = \pm e^x$$ for $$0 \leq x \leq \ln 3. If $$cross-sections perpendicular to the $$x-$$axis are squares, what is the volume of the solid?

Accepted Answer

$$16$$

Question 4

Find the volume of the solid formed by revolving the region bounded by $$y = x$$ and $$y = x^2$ about the x$-axis for 0$$ \leq x \leq 1$$.

Accepted Answer

$$\frac{2\pi}{15}$$

Question 5

Find the length of the curve $$y = \frac{e^x + e$$^{-x}$$}{2} on $$the interval $$[-\ln 3, \ln 3].$$

Accepted Answer

$$\frac{8}{3}$$

Question 6

Set up, but do not evaluate, the integral for the surface area of the solid formed by revolving $$y = 1 - x^2$ for 0$$ \leq x \leq 1$$ about the x$-axis.

Accepted Answer

$$2\pi \$$int_0$$^1 (1 - x^2$) \sqrt{1 + 4x^2$} \, dx$$

Question 7

A 10-meter chain weighing 100 kg lies on the ground. How much work is required to lift the entire chain so that the bottom end is 2 meters above the ground? (Use $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$6860 N m$$

Question 8

Evaluate the indefinite integral $$\int \ln x \, dx.$$

Accepted Answer

$$x \ln x - x + C$$

Question 9

Evaluate the definite integral $$\$$int_0$$^{\frac{\pi}{4}} 	an x \sec^2$ x \, dx.$$

Accepted Answer

$$\frac{1}{2}$$

Question 10

Evaluate the indefinite integral $$\int \frac{dx}{x^2$ + 6x + 25}.$$

Accepted Answer

$$\frac{1}{4} 	an$$^{-1}$$ \left( \frac{x+3}{4} ight) + C$$

Question 11

Evaluate the definite integral $$\$$int_1$$^e \frac{dx}{x(x+1)^2$}.$$

Accepted Answer

$$\frac{1}{2} + \frac{1}{e+1} + \ln \frac{2}{e+1}$$

Question 12

The region bounded by $$x = 1$$, $$y = 0$$, and $$y = \frac{1}{x} is $$revolved about the $$x-$$axis. The resulting volume is $$\pi \$$int_1$$^{\infty} \frac{dx}{x^2$}. $$What is the value of this improper integral?

Accepted Answer

$$1$$

Question 13

A chemical tank is being filled at a rate of $$r(t) = 100 + 20 \sin(\pi t)$$ liters per minute. If the tank starts with 1000 liters, which integral gives the total amount of water in the tank after $$T$$ minutes?

Accepted Answer

$$1000 + \$$int_0$$^T (100 + 20 \sin(\pi t)) \, dt$$

Question 14

Which of the following is the correct setup for the volume of the solid formed by revolving the region between $$y = x$$ and $$y = x^2$ about the x$-axis from x = 0$ to x = 1$?

Accepted Answer

Question 15

Suppose you want to find the work required to lift a 10-meter chain weighing 100 kg from the ground so that the bottom is 2 meters above the ground. Which integral correctly represents the work done? (Use $$g = 9.8 m/s$^2.)$$

Accepted Answer

Question 16

Which substitution is most appropriate for evaluating $$\int \frac{dx}{x^2$ + 6x + 25}$$?

Accepted Answer

Let $$u = x + 3$$, so $$x^2 + 6x + 25$$ = $$u^2 + 16$

Question 17

Which of the following integrals represents the arc length of $$y = \frac{e^x + e$$^{-x}$$}{2}$$ from $$x = -\ln 3 to$$ $$x = \ln 3$$?

Accepted Answer

$$\int_{-\ln 3}^{\ln 3} \cosh x \, dx$$

Question 18

Which of the following is the correct partial fraction decomposition for $$\frac{1}{x(x+1)^2$}$$?

Accepted Answer

$$\frac{1}{x} - \frac{1}{x+1} + \frac{1}{(x+1)^2$}$$

Question 19

Which of the following best describes the convergence of the improper integral $$\$$int_1$$^{\infty} \frac{dx}{x^2$}$$?

Accepted Answer

It converges to $$1$$

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