Evaluate the Integral

Problem

(∫_^)((x2−3*x+2)/(x+1)*d(x))

Solution

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

2. Divide x2−3*x+2 by x+1 to obtain a quotient and a remainder.

x2−3*x+2=(x−4)*(x+1)+6

3. Rewrite the integrand using the result of the division.

(x2−3*x+2)/(x+1)=x−4+6/(x+1)

4. Integrate each term separately using the power rule and the logarithmic rule for integration.

(∫_^)((x−4+6/(x+1))*d(x))=(∫_^)(x*d(x))−(∫_^)(4*d(x))+(∫_^)(6/(x+1)*d(x))

5. Apply integration rules to find the antiderivative.

(∫_^)(x*d(x))=(x2)/2

(∫_^)(4*d(x))=4*x

(∫_^)(6/(x+1)*d(x))=6*ln(x+1)

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

Final Answer

(∫_^)((x2−3*x+2)/(x+1)*d(x))=(x2)/2−4*x+6*ln(x+1)+C

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