Evaluate the Integral

Problem

(∫_^)((x3−2*x2)*(1/x−5)*d(x))

Solution

1. Expand the integrand by multiplying the two binomials using the distributive property.

(x3−2*x2)*(1/x−5)=x3*(1/x)−5*x3−2*x2*(1/x)+10*x2

2. Simplify each term in the expression before integrating.

x2−5*x3−2*x+10*x2

3. Combine like terms to reduce the number of terms in the polynomial.

−5*x3+11*x2−2*x

4. Apply the power rule for integration, which states (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term individually.

(∫_^)(−5*x3*d(x))+(∫_^)(11*x2*d(x))−(∫_^)(2*x*d(x))

5. Compute the antiderivatives and add the constant of integration C

−(5*x4)/4+(11*x3)/3−(2*x2)/2+C

6. Simplify the coefficients of the resulting expression.

−5/4*x4+11/3*x3−x2+C

Final Answer

(∫_^)((x3−2*x2)*(1/x−5)*d(x))=−5/4*x4+11/3*x3−x2+C

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