Question

Evaluate the integral. $\int \frac{d x}{x\left(x^{2}+1\right)}$

   Evaluate the integral.
$\int \frac{d x}{x\left(x^{2}+1\right)}$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 7, Problem 30

Instant Answer

Solved by Expert Alex Mangiapane

07/14/2019

Step 1

Let's set $u = x^2$. The derivative of $u$ with respect to $x$ is $du = 2x dx$. Show more…

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Evaluate the integral. $\int \frac{d x}{x\left(x^{2}+1\right)}$

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Key Concepts

Integration of Rational Functions

This concept involves techniques for evaluating integrals where the integrand is given by a ratio of polynomials. Such integrals often require methods specifically designed to handle complex rational expressions, making it a fundamental topic in calculus.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into a sum of simpler fractions. This transformation simplifies the integration process as each resulting term can usually be integrated using basic formulas, particularly when the denominator factors into linear or irreducible quadratic factors.

Algebraic Manipulation

Algebraic manipulation includes techniques such as factoring, expanding, or rewriting expressions to facilitate the application of integration methods. In the context of rational functions, it is often used to prepare the function for partial fraction decomposition by ensuring the degree of the numerator is less than that of the denominator.

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