Evaluate the Integral

Problem

(∫_^)((x4)/(x−1)*d(x))

Solution

1. Perform polynomial long division to rewrite the integrand, since the degree of the numerator is greater than the degree of the denominator.

2. Divide x4 by x−1 to obtain a quotient and a remainder.

x4=(x−1)*(x3+x2+x+1)+1

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

(∫_^)((x4)/(x−1)*d(x))=(∫_^)((x3+x2+x+1+1/(x−1))*d(x))

4. Apply the power rule for integration to the polynomial terms and the logarithmic rule to the fraction.

(∫_^)(xn*d(x))=(x(n+1))/(n+1)

(∫_^)(1/(x−1)*d(x))=ln(x−1)

5. Integrate term by term and add the constant of integration C

(x4)/4+(x3)/3+(x2)/2+x+ln(x−1)+C

Final Answer

(∫_^)((x4)/(x−1)*d(x))=(x4)/4+(x3)/3+(x2)/2+x+ln(x−1)+C

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