   Chapter 11.9, Problem 31E

Chapter
Section
Textbook Problem

# Use a power series to approximate the definite integral to six decimal places.31. ∫ 0 0.2 x ln ( 1 + x 2 )   d x

To determine

To approximate: The definite integral to six decimal places using the power series

Explanation

Result used:

(1) “The expansion of the logarithmic function log(x)=n=1(1)n1(x1)nn when 0<x2

Given:

The definite integral is 00.2xln(1+x2)dx

Calculation:

Let f(x)=00.2xln(1+x2)dx

Integrate both sides as shown below:

f(x)=00.2xln(1+x2)dx=1200.22xln(1+x2)dx

Let u=x2+1 then du=2xdx

du=2xdx12du=xdx

Substitute the values in the function f(x) .

f(x)=1200.22xln(1+x2)dx=12(0)2+1(0.2)2+1ln(u)du

By the result (1) ln(x)=n=1(1)n1(x1)nn

Show the expansion of the integral function as follows:

f(x)=1211.04n=1(1)n(u1)nndu=12[n=1(1)n1(u1)nn]du=12[n=1(1)n1(u1)n+1n(n+1)]11

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