   Chapter 7.5, Problem 59E

Chapter
Section
Textbook Problem

# Evaluate the integral.59. ∫ d x x 4 − 16

To determine

To evaluate: The integral function 1x416dx.

Explanation

Given information:

The integral function is 1x416dx.

Calculation:

Show the integral function as follows:

1x416dx (1)

Modify Equation (1) as shown below:

1x416dx=1(x2)242dx==1(x2+4)(x24)dx=1(x2+4)(x222)dx=1(x2+4)(x+2)(x2)dx (2)

Consider the function y=1(x2+4)(x+2)(x2) (3)

Expand Equation (3) using partial fraction decomposition.

1(x2+4)(x+2)(x2)=Ax+B(x2+4)+C(x+2)+D(x2)

1=(Ax+B)(x+2)(x2)+C(x2+4)(x2)+D(x2+4)(x+2) (4)

Equate the coefficients to get the value of A, B, C, and D.

Equate the coefficients of x3 in Equation (4).

A+C+D=0 (5)

Substitute 2 for x in Equation (4).

1=(Ax+B)(x+2)(x2)+C(x2+4)(x2)+D(x2+4)(x+2)=(2A+B)(2+2)(22)+C(22+4)(22)+D(22+4)(2+2)=D(4+4)×41=32DD=132 (6)

Substitute 2 for x in Equation (4).

1=(Ax+B)(x+2)(x2)+C(x2+4)(x2)+D(x2+4)(x+2)=(2A+B)(2+2)(22)+C((2)2+4)(22)+D((2)2+4)(2+2)=C(4+4)×(4)1=32CC=132 (7)

Modify Equation (5) using (6) and (7).

Substitute (132) for C and (132) for D in Equation (5)

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