Chapter 8.5, Problem 21E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 âˆ«0234x2+5x+1dx.

Formula used:

âˆ«1xdx=ln|x|+C.

Calculation:

Consider the following definite integral,

âˆ«0234x2+5x+1dx.

The above integral function can be written as,

Â 34x2+5x+1=3(x+1)(4x+1)

Now by using partial fraction method we get,

3(x+1)(4x+1)=Ax+1+B4x+1Â 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)ifÂ x=âˆ’1Â thenÂ 3=A(4(âˆ’1)+1)+B(âˆ’1+1).

3=A(âˆ’3)â†’A=âˆ’1ifÂ x=âˆ’14then3=A(4(âˆ’14)+1)+B(âˆ’14+1)3=A(âˆ’1+1)+B(34)â†’B=4

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

3(x+1)(4x+1)=âˆ’1x+1+44x+1

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