MATH 142 Spring 2015 Review Exam. Integration and Applications

Sections 5 and 6

What a great effort!

The choices of integrals are very good. There has been only one case of repetition so, except with that student, everybody else's choices made the cut. Congratulations to all of you that did basic integrals, trig integration/substitution, and especially to the team that did rational functions. All cases are contemplated, with three levels of difficulty!

The applications of integration are also really good. I miss some trigonometry, though, but the different skills needed to set up the integrals correctly are very well represented. There is one special case (the one that asks for a revolution not around a coordinate axis). I will not ask for revolution around lines other than the coordinate axes in my final, but it is good that you experiment with this problem.

The choices of improper integrals of types II are decent. I like your intermediate-level problem. The choices of improper integrals of type III are awesome. Good job! Unfortunately, the choice of type I improper integrals were not appropriate. None of them made the cut. By the way, I will ask something of intermediate level in the test, so make sure to learn the skills needed to identify the "improperness," and develop the correct treatment.

  1. Compute the following (basic) integrals. Note that in some cases you may need to use substitution, or techniques based upon “integration by parts.” The key of these questions is to be able to identify quickly the technique of integration.
    \begin{align} & \int \Big( \frac{\ln x}{x} \Big)^2 \, dx & & \int \frac{x^3}{(x^2+5)^2}\, dx & & \int x^3 ( \sin^2 x -1)\, dx \\ & \int (x-25)^2 e^{2x}\, dx & & \int \frac{12\pi}{5\sqrt{1-x^2}}\, dx & & \int \frac{7\pi + 2x^2}{5x^3}\, dx \\ & \int -7 \csc^2 x e^{\cot x}\, dx & & \int -3e^x \sec^2 (e^x)\, dx & & \int (e^x +23^x-\sin x)\, dx \\ & \int \csc x \cot x\, dx & & \int x^7 + \frac{3}{x^2}\, dx & & \int \frac{\cos(5x)}{e^{\sin(5x)}}\, dx \\ & \int \frac{\tan^{-1}(4x)}{1+16x^2}\, dx & & \int \frac{12x-28}{3x^2-14x+7}\, dx & & \int \big( \cos^3 x + 3\cos^2 x \big) \sin x\, dx \end{align}
  2. Compute the area of the regions delimited by the curves of the following functions:
    \begin{align} & (a) \quad y=x^2, x=2y \\ & (b) \quad y=(x-1)^3, y=x-1 \\ & (c) \quad y=5-2x^2, y=3x^2, x=-4, x=4 \\ & (d) \quad x=2y, x=y-y^2 \\ & (e) \quad y=x^3-14x, y=2x \\ & (f) \quad y=2x^2, y=3x-x^2 \end{align}
  3. Compute the volume of the solids of revolution obtained by rotating the region bounded by the following curves around the specified axes:
    \begin{align} & (a) \quad y=x^2-4x+5, x=1, x=4, \text{ about the }x\text{-axis.} \\ & (b) \quad y=x^2-2x, y=x, \text{ about the line }y=4. \\ & (c) \quad y=5x^3, y=5x, x\geq 0, \text{ about the }x\text{-axis}. \\ & (d) \quad y=x^2/9, x=3, y=0, \text{ about the }y\text{-axis.} \\ & (e) \quad x=4y^2-y^3, \text{about the }y\text{-axis.} \\ & (f) \quad 4y=x^2, x=3, y=0, \text{about the }y\text{-axis.} \end{align}
  4. The following fall into the categories of “Trigonometric Integration,” and “Trigonometric substitution.” Compute the following integrals.
    \begin{align} & \int \frac{x^3}{\sqrt{25+x^2}}\, dx & & \int \frac{dx}{\sqrt{x^2-4}} & & \int \frac{dx}{x^2\sqrt{x^2+6}} \\ & \int \sin(10x) \cos(7x)\, dx & & \int \sin^5 x \cos^2 x\, dx & & \int \frac{9x}{(1-x^2)^{3/2}}\, dx \\ & \int \sin^2 x \cos^2 x\, dx & & \int \csc x\, dx & & \int \tan^3 x \sec^3 x\, dx \\ & \int \sin^2 x \, dx & & \int \cot^2 x\, dx \end{align}
  5. The following fall into the category of “Integration of Rational Functions.” Compute the following:
    \begin{align} & \int \frac{3r^2}{r+9}\, dr & & \int \frac{3x^2+2x-3}{x(x^2-1)}\, dx & & \int \frac{7x^2+x+7}{(x^2+1)^2}\, dx \\ & \int \frac{x^3+8x-7}{x+4}\, dx & & \int \frac{dx}{2x^2-12x+26} & & \int \frac{x^5+2}{x^2-2}\, dx \\ & \int \frac{x^3+2x-1}{(x^2-x-2)^2}\, dx & & \int \frac{4x^2-3x+2}{4x^2-4x+3}\, dx & & \int \frac{3x-1}{x^2+10x+28}\, dx \\ & \int \frac{x^4-5x^3+6x^2-18}{x^3-3x^2}\, dx \end{align}
  6. Find a partial fraction decomposition of the functions below (do not solve for the variables):
    \begin{align} f(x) &= \frac{5x+4}{(x+4)^3 (x^2+5)^2}, & g(x) &= \frac{4}{x^6-6x^3}. \end{align}
  7. Compute the following:
    \begin{align} & \int_0^1 \frac{4}{x^3}\, dx & & \int_1^6 \frac{dx}{(x-1)^3} & & \int_2^8 \frac{5}{x^3-8}\, dx \\ & \int_{-1}^1 \frac{dx}{x} & & \int_{-\pi/2}^{\pi/2} \csc x \cot x\, dx & & \int_1^\infty \frac{dx}{2-x} \end{align}