Evaluate the Integral

Problem

(∫_^)((5*x5+5*x2+10)/(x3−x)*d(x))

Solution

1. Perform polynomial long division because the degree of the numerator is greater than the degree of the denominator.

5*x5+5*x2+10=(5*x2+5)*(x3−x)+(5*x2+5*x+10)

2. Rewrite the integral using the result of the division.

(∫_^)((5*x5+5*x2+10)/(x3−x)*d(x))=(∫_^)((5*x2+5+(5*x2+5*x+10)/(x3−x))*d(x))

3. Factor the denominator of the remaining fraction.

x3−x=x*(x2−1)=x*(x−1)*(x+1)

4. Apply partial fraction decomposition to the rational part.

(5*x2+5*x+10)/(x*(x−1)*(x+1))=A/x+B/(x−1)+C/(x+1)

5. Solve for the constants A B and C by clearing the denominator.

5*x2+5*x+10=A*(x−1)*(x+1)+B*x*(x+1)+C*x*(x−1)

6. Substitute values for x to find the coefficients.

Let *x=0⇒10=A*(−1)⇒A=−10

Let *x=1⇒20=B(2)⇒B=10

Let *x=−1⇒10=C(2)⇒C=5

7. Substitute the partial fractions back into the integral.

(∫_^)((5*x2+5−10/x+10/(x−1)+5/(x+1))*d(x))

8. Integrate term by term using the power rule and the natural logarithm rule.

(5*x3)/3+5*x−10*ln(x)+10*ln(x−1)+5*ln(x+1)+C

Final Answer

(∫_^)((5*x5+5*x2+10)/(x3−x)*d(x))=(5*x3)/3+5*x−10*ln(x)+10*ln(x−1)+5*ln(x+1)+C

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