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

Guida verificata passo dopo passo

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.

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Polynomial Long Division

When the degree of the numerator is equal to or greater than the denominator in a rational function, polynomial long division is used to simplify the integrand. This process rewrites the integrand as a polynomial plus a proper fraction, making the integral easier to evaluate.

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Integration of Rational Functions

Integrating rational functions often involves breaking them into simpler parts, such as polynomials and proper fractions. After simplification, standard integration techniques like substitution or partial fractions can be applied to evaluate the integral.

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Recognizing Perfect Square Trinomials

Identifying perfect square trinomials in the denominator, such as (x - 5)², helps simplify the integral. This recognition allows substitution methods or rewriting the integrand in a form that is easier to integrate.

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