# Integration by parts, Partial fraction expansion, Improper Integrals

• MHB
• ertagon2
In summary, the conversation revolves around mathematical calculations and limit functions. The participants discuss the correct answer to a question, the properties of an odd function, and the simplification of a fraction. They also point out a mistake in the calculation of some variables and explain the simplification of a limit function.
ertagon2
View attachment 7785
1. -
2. check if right
3. check if right
4. Now, 2 seems to be the right answer for A yet when i made x=5 and subtracted new form form the old one I got a difference of ~$\frac{4}{9}$ (should be 0 obviously) I got A=2 B=$\frac{45}{21}$ C=2
5. How to calculate $\lim_{{x}\to{\infty}}(- e^{-x})$

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2. Correct
3. Yes, notice that it's an odd function $f(-x)=-f(x)$ so the "areas" cancel out.
4. Might want to check your math here a bit. The answer is correct ($A=2$), but $B=3$ and $C=-2$.
Also, $\lim_{{x}\to{\infty}}(e^{-x})=\lim_{{x}\to{\infty}}1/e^x=0$ since the denominator goes to infinity.

Since when is 0.25 a fraction in simplest form?

Prove It said:
Since when is 0.25 a fraction in simplest form?

so =$\frac{1}{4}$ ?

## 1. What is integration by parts?

Integration by parts is a technique used in calculus to evaluate the integral of a product of two functions. It is based on the product rule for differentiation, and involves choosing one function to be differentiated and the other to be integrated.

## 2. How does partial fraction expansion work?

Partial fraction expansion is a method used to simplify and solve integrals involving rational functions. It involves breaking down a fraction into simpler fractions with known forms, and then integrating each of these simpler fractions separately.

## 3. What are improper integrals?

Improper integrals are integrals that do not have a finite value. This can occur when the function being integrated is unbounded or undefined at certain points, or when the interval of integration is infinite.

## 4. How do you determine the convergence or divergence of an improper integral?

To determine the convergence or divergence of an improper integral, you can use the comparison test, limit comparison test, or the p-test. These tests involve comparing the integral to a known convergent or divergent series or function.

## 5. Can improper integrals be solved using traditional integration techniques?

Yes, improper integrals can be solved using traditional integration techniques, such as integration by parts or partial fraction expansion. However, it is important to take into account the convergence or divergence of the integral in order to obtain an accurate result.

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