Question

Evaluate the integral. $\int_0^{1 / 2} x \cos \pi x d x$

   Evaluate the integral.
 $\int_0^{1 / 2} x \cos \pi x d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 23

Instant Answer

Solved by Expert Wen Zheng

01/31/2018

Step 1

The integral given is \(\int_0^{1/2} x \cos(\pi x) \, dx\). This is a definite integral involving a product of a polynomial and a trigonometric function, which suggests the use of integration by parts. Show more…

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Evaluate the integral. $\int_0^{1 / 2} x \cos \pi x d x$

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Key Concepts

Definite Integral

A definite integral computes the accumulation of a quantity, such as area under a curve, between two specified limits. It involves finding an antiderivative of the function and evaluating the result at the upper and lower limits using the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

This theorem bridges the concepts of differentiation and integration by stating that if a function is continuous over an interval, then its definite integral over that interval can be computed using any of its antiderivatives. It allows for the evaluation of a definite integral by subtracting the antiderivative's value at the lower limit from its value at the upper limit.

Integration of Trigonometric Functions

Integrating trigonometric functions, such as cosine, involves applying standard antiderivatives. For cosine in particular, the antiderivative is the sine function. Understanding how to integrate trigonometric functions is key in many areas of analysis and physics.

Constant Multiple Rule in Integration

This rule states that if a constant multiplies a function inside an integral, the constant can be factored out of the integral. This property simplifies the integration process by reducing the problem to finding the antiderivative of the remaining function, and then multiplying the result by the constant.

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