Calculus: Integration Techniques
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Questions and Answers

What is the integral of $rac{1}{x-1} + rac{1}{(x-1)^2}$ with respect to $x$?

  • $rac{ ext{ln}(x-1)}{2} - rac{1}{(x-1)^2}$
  • $rac{1}{2 ext{ln}|x-1|} + rac{1}{(x-1)^2}$
  • $ ext{ln}|x-1| - rac{1}{x-1}$ (correct)
  • $rac{1}{2} ext{ln}|x-1|$

    • What is the integral of $ an(p) ext{sec}^2(p)$ with respect to $p$?

  • $rac{1}{2} ext{ln}|p|$
  • $rac{1}{2} ext{sec}(p)$
  • $rac{1}{2} ext{ln}| ext{sec}(p)|$
  • $- ext{ln}| ext{cos}(p)|$ (correct)

    • Which of the following is a correct step when solving $rac{1}{x-1} + rac{1}{(x-1)^2}$?

  • Rewrite it as $rac{d}{dx}((x-1)^{-1})$
  • Rewrite it as $rac{d}{dx} ext{ln}|x-1|$ (correct)
  • Combine like terms first
  • Use integration by parts

    • What technique is primarily used to evaluate $ an(p) ext{sec}^2(p)$?

    <p>u-substitution</p> Signup and view all the answers

    How can $rac{1}{(x-1)^2}$ in the integral be treated to simplify the calculation?

    <p>As the derivative of $(x-1)^{-1}$</p> Signup and view all the answers

    Study Notes

    Integration of Rational Functions

    • The integral of $\frac{1}{x-1} + \frac{1}{(x-1)^2}$ with respect to $x$ involves evaluating two separate terms.

    Trigonometric Integration

    • The integral of $\tan(p) \sec^2(p)$ with respect to $p$ is a classic example of a trigonometric integration problem.
    • This type of integral can be evaluated using the substitution technique.

    Solving Rational Functions

    • When solving the integral of $\frac{1}{x-1} + \frac{1}{(x-1)^2}$, a correct step is to split the fraction into two separate terms.
    • This allows for the application of different integration techniques to each term.

    Integration Techniques

    • The integral of $\tan(p) \sec^2(p)$ primarily uses the substitution technique for evaluation.
    • The substitution technique involves replacing the original function with a more manageable expression.

    Simplifying Integration

    • The term $\frac{1}{(x-1)^2}$ in the integral can be treated as a derivative of the first term, $\frac{1}{x-1}$.
    • Recognizing this relationship allows for the simplification of the calculation and the application of a more straightforward integration technique.

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    Description

    Solve integrals involving rational functions and trigonometric functions. Practice integration by parts, partial fractions, and substitution methods.

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