   Chapter 8.5, Problem 21E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral In Exercises 21-24, use partial fractions to evaluate the definite integral. Use a graphing utility to verify your result. ∫ 0 2 3 4 x 2 + 5 x + 1 d x .

To determine
The value of given definite integral by using partial fractions.

Explanation

Given:

The definite integral is 0234x2+5x+1dx.

Formula used:

1xdx=ln|x|+C.

Calculation:

Consider the following definite integral,

0234x2+5x+1dx.

The above integral function can be written as,

34x2+5x+1=3(x+1)(4x+1)

Now by using partial fraction method we get,

3(x+1)(4x+1)=Ax+1+B4x+1 34x2+5x+1=A(4x+1)+B(x+1)(x+1)(4x+1)

By simplifying further we get,

3=A(4x+1)+B(x+1)if x=1 then 3=A(4(1)+1)+B(1+1).

3=A(3)A=1if x=14then3=A(4(14)+1)+B(14+1)3=A(1+1)+B(34)B=4

So by substituting the value of A and B we get,

3(x+1)(4x+1)=1x+1+44x+1

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