Math 152: Worksheet 10

Strategies for Integration

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Evaluate $\displaystyle \int \frac{dx}{x(x-1)^2} $

• Hint 1

  This integral should be done by Partial Fractions.

• Hint 2

  The decomposition you want is $\displaystyle \frac{A}{x} + \frac{B}{x-1} + \frac{C}{(x-1)^2}$

• Hint 3

  The function you need to integrate is $\displaystyle \frac{1}{x} - \frac{1}{x-1} + \frac{1}{(x-1)^2}$

• Hint 4

  Once you're done, use a triangle to switch this back to being in terms of $x$.

• Answer

  $\displaystyle \ln{|x|} - \ln{|x-1|} - \frac{1}{x-1} + C$

Problem 2

Evaluate $\displaystyle \int \frac{x}{(x^2 - 1)^{3/2}}\ dx $

• Hint 1

  This is just a substitution problem.

• Hint 2

  Set $u = x^2 - 1$, so that $du = 2x\ dx$.

• Answer

  $\displaystyle \frac{1}{\sqrt{x^2 - 1}} + C$

Problem 3

Compute $\displaystyle \int \frac{x^3}{(x^2 - 1)^{3/2}}\ dx$

• Hint 1

  This can be done with either trigonometric substitution or normal substitution.

• Hint 2

  For trigonometric substitution, set $x = sec(\theta)$. For normal substitution, set $u = x^2 - 1$ and manipulate whatever is left in the numerator to be in terms of $u$.

• Answer

  $\displaystyle \sqrt{x^2 - 1} - \frac{1}{\sqrt{x^2 - 1}} + C$

Problem 4

Compute $\displaystyle \int \frac{x^4 + 1}{x^2 + 1} \ dx$

• Hint 1

  There are a few ways to handle this integral here. The easiest way might be long division into using known integral formulas.

• Hint 2

  Long division gives that $\displaystyle \frac{x^4 + 1}{x^2 + 1} = x^2 - 1 + \frac{2}{x^2 + 1}$

• Answer

  $\displaystyle \frac{x^3}{3} - x + 2 \tan^{-1}(x) + C$

Problem 5

Compute $\displaystyle \int \tan{x} \sec^{5/4}{x}\ dx$

• Hint 1

  This is related to both trigonometric integrals and substitution.

• Hint 2

  What choice could I make for $u$ and $du$ here to make this easy? Look for $du$.

• Hint 3

  Take $du = \sec{x} \tan{x}$, which comes from $u = \sec{x}$.

• Hint 4

  This integral then becomes $\displaystyle \int u^{1/4}\ du$

• Answer

  $\displaystyle \frac{4}{5} \sec^{5/4}(x)+ C$

Problem 6

Compute $\displaystyle \int e^x \sqrt{e^{2x} - 1}\ dx $. Take your time on this one. It's going to need many different techniques.

• Hint 1

  Your best bet here is to start with a substitution, and see where that gets you.

• Hint 2

  Try $u = e^x$ and see what is left, which should leave you with needing to integrate $\sqrt{u^2 - 1}$, which is a trigonometric substitution.

• Hint 3

  After this, you'll get to integrating $\sec{\theta} \tan^{2}(\theta)$, which requires trigonometric integrals.

• Hint 4

  This is the same as integrating $\sec^3(\theta) - \sec(\theta)$, which needs reduction, and then you need to convert back to $x$.

• Answer

  $\displaystyle \frac{1}{2}(e^x \sqrt{e^{2x} - 1} - \frac{1}{2} \ln|e^x + \sqrt{e^{2x} - 1}| + C$

Submission Problems

Problem 1

Compute $\displaystyle \int x \sec^{-1}{x}\ dx$.

Problem 2

Compute $\displaystyle \int \sqrt{x^2 + 6x}\ dx $