Question

23-64. Integration Evaluate the following integrals.47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

Accepted Answer

First, observe the integral $$ \int \frac{x^{3} - 10x^{2} + 27x}{x^{2} - 10x + 25} \, dx . $$Notice that the denominator $$ x^{2} - 10x + 25 $$ can be factored or recognized as a perfect square. Specifically, $$ x^{2} - 10x + 25 = (x - 5)^{2} . $$Next, perform polynomial division because the degree of the numerator (3) is higher than the degree of the denominator (2). Divide $$ x^{3} - 10x^{2} + 27x  by$$ $$ (x - 5)^{2}  to $$rewrite the integrand as a polynomial plus a proper rational function. After the division, express the integrand as $$ Q(x) + \frac{R(x)}{(x - 5)^{2}} $$, where $$ Q(x)  is $$the quotient polynomial and $$ R(x)  is $$the remainder polynomial with degree less than 2. Then, split the integral into two parts: $$ \int Q(x) \, dx + \int \frac{R(x)}{(x - 5)^{2}} \, dx . $$The first integral is straightforward since $$ Q(x)  is a $$polynomial. For the second integral, use substitution $$ u = x - 5  to $$simplify the denominator and then integrate the resulting rational function by breaking it into simpler terms if necessary.

23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

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Key Concepts

Polynomial Long Division

Integration of Rational Functions

Recognizing Perfect Square Trinomials