Evaluate the Integral

Problem

(∫_^)((x2)/(x−1)*d(x))

Solution

1. Perform polynomial long division to rewrite the integrand because the degree of the numerator is greater than or equal to the degree of the denominator.

x2=(x−1)*(x+1)+1

2. Rewrite the fraction by dividing each term in the numerator by the denominator.

(x2)/(x−1)=x+1+1/(x−1)

3. Apply the sum rule for integration to integrate each term individually.

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

4. Integrate each term using the power rule for x and 1 and the natural logarithm rule for the fraction.

(∫_^)(x*d(x))=(x2)/2

(∫_^)(1*d(x))=x

(∫_^)(1/(x−1)*d(x))=ln(x−1)

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

(x2)/2+x+ln(x−1)+C

Final Answer

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

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