Evaluate the Integral

Problem

(∫_^)((x2−1)/(x3+x)*d(x))

Solution

1. Factor the denominator to prepare for partial fraction decomposition.

x3+x=x*(x2+1)

2. Set up the partial fraction decomposition for the integrand.

(x2−1)/(x*(x2+1))=A/x+(B*x+C)/(x2+1)

3. Solve for the constants by multiplying both sides by the common denominator.

x2−1=A*(x2+1)+(B*x+C)*x

4. Expand and group the terms to compare coefficients.

x2−1=(A+B)*x2+C*x+A

5. Equate the coefficients to find the values of A B and C

A=−1

C=0

A+B=1⇒−1+B=1⇒B=2

6. Rewrite the integral using the decomposed fractions.

(∫_^)((x2−1)/(x3+x)*d(x))=(∫_^)(((−1)/x+(2*x)/(x2+1))*d(x))

7. Integrate each term separately using the natural logarithm rule and substitution.

(∫_^)((−1)/x*d(x))=−ln(x)

(∫_^)((2*x)/(x2+1)*d(x))=ln(x2+1)

8. Combine the results and add the constant of integration C

ln(x2+1)−ln(x)+C=ln((x2+1)/x)+C

Final Answer

(∫_^)((x2−1)/(x3+x)*d(x))=ln((x2+1)/x)+C

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