Math Problem Statement

∫▒x/〖(x^2-1)〗^2 dx=

Solution

To solve the integral x(x21)2dx\int \frac{x}{(x^2-1)^2} \, dx, we can use the substitution method. Let's go through the steps:

  1. Substitution: Let u=x21u = x^2 - 1. Then, du=2xdxdu = 2x \, dx or du2=xdx\frac{du}{2} = x \, dx.

  2. Rewrite the integral: Substituting uu and dudu into the integral: x(x21)2dx=xu2du2x\int \frac{x}{(x^2-1)^2} \, dx = \int \frac{x}{u^2} \cdot \frac{du}{2x} The xx terms cancel out: 12u2du\int \frac{1}{2u^2} \, du

  3. Integrate: The integral now is: 12u2du=12u2du\int \frac{1}{2u^2} \, du = \frac{1}{2} \int u^{-2} \, du Using the power rule for integration undu=un+1n+1+C\int u^n \, du = \frac{u^{n+1}}{n+1} + C, we get: 12u2du=12(u11)+C=12u+C\frac{1}{2} \int u^{-2} \, du = \frac{1}{2} \left( \frac{u^{-1}}{-1} \right) + C = -\frac{1}{2u} + C

  4. Substitute back: Replace uu with x21x^2 - 1: 12u+C=12(x21)+C-\frac{1}{2u} + C = -\frac{1}{2(x^2 - 1)} + C

So, the solution to the integral is: x(x21)2dx=12(x21)+C\int \frac{x}{(x^2-1)^2} \, dx = -\frac{1}{2(x^2 - 1)} + C

Would you like any further details or have any questions?

Here are 5 questions you might want to ask next:

  1. How do you solve a definite integral for a similar function?
  2. Can you explain the substitution method in more detail?
  3. How do you handle integrals with different forms of rational functions?
  4. What are some common techniques for integrating rational functions?
  5. Can you provide examples of more complex integrals and their solutions?

Tip: Always check if substitution or partial fraction decomposition can simplify the integration process.

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method
Rational Functions

Formulas

Integration by substitution

Theorems

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Suitable Grade Level

Advanced High School and Above