   Chapter 8.2, Problem 50E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral In Exercises 43-52, evaluate the definite integral. Use a graphing utility to verify your result. ∫ 0 1 ln ( 4 + x 2 ) d x

To determine

To calculate: To find the value of definite integral

01ln(4+x2)dx. And also verify it by using graphing utility.

Explanation

Given: the given definite integral is 01ln(4+x2)dx

Formula used:

By using Integral by part i.e.,

udv=uvvdu

Calculation:

The given integration is

01ln(4+x2)dx

To Find integration by part method,

Let u=ln(4+x2)

On Differentiating with respect to x

ddxu=ddxln(4+x2)dudy=2x(4+x2)

And

dv=dx

Now, on integrating both sides

we get,

v=x

Recall the formula of integral by parts i.e.

udv=uvvdu

On Substituting the values above

we get the integral,

ln(4+x2)=xln(4+x2)2x(4+x2)×xdx=xln(4+x2)2x2(4+x2)dx=xln(4+x2)4[tan1(x2)+x2]+C=xln(4+x2)+4tan1(x2)2x+C

Now, by using antiderivative to find the value of integral

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