Question

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

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

Calculus: Early Transcendental Functions

Calculus: Early Transcendental Functions

Robert T. Smith,… 3rd Edition

Chapter 6, Problem 31

Instant Answer

Solved by Expert Audrey Fong

06/03/2020

Step 1

Step 1: First, we separate the integral into two parts: $$\int \frac{1+x}{1+x^{2}} d x = \int \frac{1}{1+x^{2}} d x + \int \frac{x}{1+x^{2}} d x$$ Show more…

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

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Key Concepts

Integration of Rational Functions

This concept involves integrating functions that are ratios of polynomials. Approaches can often include algebraic manipulation, such as decomposing the rational function into simpler parts, or applying substitution techniques. In many cases, understanding the degree of the numerator relative to the denominator helps decide whether to perform polynomial long division or to split the integrals.

Splitting the Integrand

In many integrals involving a sum or difference in the numerator, it is beneficial to decompose the integrand into separate, simpler fractions. This technique allows the integral to be expressed as a sum of integrals, each of which can be handled with standard methods, such as substitution or using standard integral formulas.

u-Substitution

u-Substitution is a technique for simplifying integrals by substituting part of the integrand with a new variable. This method is particularly effective when the derivative of the chosen substitution appears elsewhere in the integrand. It transforms the integral into a form that is easier to solve, often leading to expressions like logarithms.

Standard Integral Forms

Many integrals rely on well-known antiderivatives. Two important standard forms are the inverse trigonometric form, where the integral of 1/(1+x^2) leads to the inverse tangent function, and the logarithmic form, which arises in integrals of the type x/(1+x^2), where substitution yields a natural logarithm. Familiarity with these standard results is essential for efficient integration.

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