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Questions and Answers
What is the integral of $rac{1}{x-1} + rac{1}{(x-1)^2}$ with respect to $x$?
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 $p$?
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 $rac{1}{x-1} + rac{1}{(x-1)^2}$?
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)$?
What technique is primarily used to evaluate $ an(p) ext{sec}^2(p)$?
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How can $rac{1}{(x-1)^2}$ in the integral be treated to simplify the calculation?
How can $rac{1}{(x-1)^2}$ in the integral be treated to simplify the calculation?
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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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