   Chapter 8.5, Problem 34E

Chapter
Section
Textbook Problem

Verifying a Formula In Exercises 33-36, use the method of partial fractions to verify the integration formula. ∫ 1 a 2 − x 2   d x = 1 2 a ln | a + x a − x | + C

To determine

To prove: The solution of the integral 1a2x2dx=12aln|a+xax|+C specified using partial fractions.

Explanation

Given:

The integral is 1a2x2dx=12aln|a+xax|+C.

Proof:

Consider the specified integral:

1a2x2dx

Use partial fractions express the integrant as follow:

1a2x2=1(a+x)(ax)=Aa+x+Bax1=A(ax)+B(a+x)

Substitute x=a as follows:

1=B(2a)B=12a

Substitute x=a as follows:

1=A(2a)A=12a

Now, solve as shown below:

1a2x2dx=

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