Question

Evaluate the integral. $\int(x-1) \sin \pi x d x$

   Evaluate the integral.
$\int(x-1) \sin \pi x d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 6

Instant Answer

Solved by Expert Danjoseph Quijada

12/23/2017

Step 1

The integral \(\int (x-1) \sin \pi x \, dx\) is a product of a polynomial and a trigonometric function, which suggests using integration by parts. Show more…

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Evaluate the integral. $\int(x-1) \sin \pi x d x$

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Key Concepts

Integration by Parts

Integration by parts is a fundamental technique used to integrate products of functions. It is based on the formula ? u dv = uv - ? v du, which originates from the product rule for differentiation. This method is especially useful when one of the functions in the product simplifies upon differentiation, enabling the overall integral to be more easily evaluated.

Integration of Trigonometric Functions

Integrating trigonometric functions, particularly when they involve a linear function of the variable (like a sine or cosine with a multiple of x), relies on standard antiderivative formulas. For example, one key formula is the antiderivative of sin(ax), which is - (1/a) cos(ax). Knowledge of these standard integrals is essential for handling integrals that mix polynomials with trigonometric functions.

Linearity of Integration

The linearity of integration is the principle that the integral of a sum or difference of functions is the sum or difference of the integrals of those functions. This property allows the decomposition of complex integrals into simpler parts, each of which can be integrated separately, greatly simplifying the process of finding antiderivatives.

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