Indefinite integral in division form

• MHB
• Elina_Gilbert
In summary, the conversation discusses the steps taken to solve the integral $\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx$ and the struggle to find a solution. Despite using various substitutions, the integral remains unsolvable. The speaker also asks for suggestions on alternative methods to solve the integral.
Elina_Gilbert
I have the following integration -

$$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx$$

To solve this I did the following -
$$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx$$

Which gives me -

$$log(x) + C+ \int \frac{1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx$$

No matter what substitution I do, I couldn't solve the integral -

$$\int \frac{1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx$$

Can anyone please suggest what I did wrong? Please suggest me another method to solve this?

May I ask what context this integral comes from?

What is an indefinite integral in division form?

An indefinite integral in division form is a mathematical expression that represents the antiderivative or the reverse process of differentiation. It is written in the form of ∫f(x)/g(x) dx, where f(x) and g(x) are functions and dx represents the variable of integration.

What is the purpose of calculating an indefinite integral in division form?

The purpose of calculating an indefinite integral in division form is to find the original function or the function that, when differentiated, gives the given function in the integral. It is a fundamental tool in calculus and is used to solve various problems in physics, engineering, and other fields.

How is an indefinite integral in division form different from a definite integral?

An indefinite integral in division form has no limits of integration, whereas a definite integral has specific limits of integration. This means that the result of an indefinite integral is a function, while the result of a definite integral is a numerical value.

What are some common techniques for solving indefinite integrals in division form?

Some common techniques for solving indefinite integrals in division form include substitution, integration by parts, and partial fractions. These techniques involve manipulating the integral to make it easier to solve, using known formulas and rules of integration.

Why is it important to understand indefinite integrals in division form?

Understanding indefinite integrals in division form is crucial for solving various problems in mathematics and other fields. It allows us to find the original function from a given derivative, which is essential in understanding the behavior and properties of functions. It also plays a significant role in the development of more advanced mathematical concepts and theories.

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