   Chapter 8.5, Problem 17E

Chapter
Section
Textbook Problem

# Using Partial Fractions In Exercises 5–22, use partial fractions to find the indefinite integral. ∫ x 2 − 1 x 3 + x

To determine

To calculate: The indefinite integral x21x3+xdx by the use of partial fractions.

Explanation

Given:

The provided integral is x21x3+xdx.

Formula used:

Power rule of integration is;

xndx=xn+1n+1+C

Log rule of integration is;

1xdx=ln|x|+C

Calculation:

Consider the integral x21x3+xdx.

Rewrite the denominator in factors which cannot be simplified further as;

x3+x=x(x2+1)

Now include one partial fraction of each power of xandx2+1 as;

x21x(x2+1)=Ax+Bx+Cx2+1

Multiply the least common basic denominator x(x2+1) which gives the basic equation as;

x21=Axx(x2+1)+Bx+Cx2+1x(x2+1)=A(x2+1)+(Bx+C)x

Solve the basic equation for x=0 to find the value of A as;

x21=A(x2+1)+(Bx+C)x01=A(0+1)+01=A

Substitute the derived value of A in basic equation as;

x21=A(x2+1)+(Bx+C)xx21=1(x2+1)+(Bx+C)xx21=x21+(Bx+C)x2x2=(Bx+C)x

Simplify further as;

2x2=(Bx+C)xBx+C=2x

Consider the equation:

Bx+C=2x

Put x=1 in Bx+C=2x to get equation the equation as;

Bx+C=2xB1+C=21B+C=2

Put x=1 in Bx+C=2x to get equation as;

Bx+C=2x

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