   Chapter 7.1, Problem 67E

Chapter
Section
Textbook Problem

# The Fresnel function S ( x ) = ∫ 0 x sin ( 1 2 π t 2 ) d t was discussed in Example 4.3.3 and is used extensively in the theory of optics. Find ∫ S ( x ) d x . [Your answer will involve S ( x ) . ]

To determine

To evaluate: the given integral

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts is given by

udv=uvvdu

Given:

The Fresnel function, S(x)=0xsin(12πt2)dt

Calculation:

Consider the following integral:

S(x)dx

Solve the integral using integration by parts. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=S(x)      dv=dx

Then, the differentiation of u and antiderivative of dv will be

du=ddxS(x)     v=x

Recall that by fundamental theorem of calculus,ddxaxf(t)dt=f(x). Hence, the differential du will be:

du=ddx0xsin(12πt2)dt=sin(12πx2)

Integration by parts gives the integration as:

S(x)

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