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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.1.32
Chapter 8, Problem 8.1.32

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁¹ (√(1 + x²) sin x) dx

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1
First, examine the integrand \(\sqrt{1 + x^{2}} \sin x\) to determine if it has any symmetry properties. Note that \(\sqrt{1 + x^{2}}\) is an even function because replacing \(x\) by \(-x\) does not change its value, while \(\sin x\) is an odd function because \(\sin(-x) = -\sin x\).
Since the integrand is the product of an even function and an odd function, the overall integrand is an odd function. Recall that the product of an even function and an odd function is an odd function.
The integral of an odd function over a symmetric interval \([-a, a]\) is zero. This is because the areas on the negative and positive sides cancel each other out.
Therefore, without performing any algebraic manipulation or substitution, conclude that the value of the integral \(\int_{-1}^{1} \sqrt{1 + x^{2}} \sin x \, dx\) is zero due to the odd symmetry of the integrand.
If you want to verify this result, you could split the integral into two parts from \(-1\) to \(0\) and from \(0\) to \(1\) and check that they are negatives of each other, confirming the integral sums to zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals and Symmetry

Definite integrals calculate the net area under a curve between two limits. When the integrand involves symmetric intervals, analyzing whether the function is even, odd, or neither can simplify evaluation. For example, the integral of an odd function over [-a, a] is zero.
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Properties of Even and Odd Functions

A function f(x) is even if f(-x) = f(x) and odd if f(-x) = -f(x). The product of an even and an odd function is odd, and the product of two even or two odd functions is even. Recognizing these properties helps determine integral values over symmetric intervals.
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Integration Techniques: Substitution and Algebraic Manipulation

When direct integration is difficult, substitution or algebraic manipulation can simplify the integral. Substitution changes variables to transform the integral into a more manageable form, while algebraic methods or trigonometric identities can rewrite the integrand for easier integration.
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