Evaluate the Integral

Problem

(∫_^)((2*x2+7*x−3)/(x−2)*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 2*x2+7*x−3 by x−2 to get a quotient and a remainder.

2*x2+7*x−3=(2*x+11)*(x−2)+19

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

(2*x2+7*x−3)/(x−2)=2*x+11+19/(x−2)

4. Integrate term by term using the power rule for 2*x and 11 and the logarithmic rule for the fraction.

(∫_^)((2*x+11+19/(x−2))*d(x))=(∫_^)(2*x*d(x))+(∫_^)(11*d(x))+(∫_^)(19/(x−2)*d(x))

5. Apply integration rules to find the antiderivative.

(∫_^)(2*x*d(x))=x2

(∫_^)(11*d(x))=11*x

(∫_^)(19/(x−2)*d(x))=19*ln(x−2)

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

Final Answer

(∫_^)((2*x2+7*x−3)/(x−2)*d(x))=x2+11*x+19*ln(x−2)+C

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