Question

Evaluate each integral. $$\int_{-1}^{1} \frac{1}{1+x^{2}} d x$$

   Evaluate each integral.
$$\int_{-1}^{1} \frac{1}{1+x^{2}} d x$$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 6, Problem 63

Instant Answer

Solved by Expert Rukhmani Jain

06/13/2021

Step 1

This is because the derivative of $\tan^{-1}(x)$ is $\frac{1}{1+x^{2}}$. Show more…

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Evaluate each integral. $$\int_{-1}^{1} \frac{1}{1+x^{2}} d x$$

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Key Concepts

Antiderivatives and the Fundamental Theorem of Calculus

This concept involves finding an antiderivative of a function and then applying the Fundamental Theorem of Calculus, which relates the antiderivative at the boundary points to the definite integral. In the context of the integral provided, recognizing the antiderivative of 1/(1+x²) as arctan(x) allows for straightforward evaluation.

Standard Integration Techniques for Rational Functions

Certain rational functions, like 1/(1+x²), have known standard integrals. Recognizing these allows one to use established integral formulas, such as the fact that ?(1/(1+x²)) dx equals arctan(x) plus a constant.

Properties of Even Functions in Definite Integration

An even function satisfies f(x) = f(-x), which means that the definite integral over symmetric limits about zero can be simplified by doubling the integral from 0 to the upper limit. This property simplifies the computation of the given integral when the function involved is even.

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