Question

78. Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).e. ∫ (2x² - 3x) / (x - 1)³ dx

Accepted Answer

First, simplify the integrand by performing polynomial division of the numerator $$2x^{2} - 3x$ by the denominator (x$$ - $$1)^{3}. $$Since the denominator is a cubic term, consider rewriting the integrand to a form suitable for integration or partial fractions if possible. Rewrite the integral as $$\int \frac{2x$$^{2}$$ - 3x}{(x - 1$$)^{3}$$} \, dx$$ and consider substituting $$u = x - 1 to $$simplify the denominator. Then express the numerator in terms of $$u as $$well. After substitution, express the integrand as a sum of terms with powers of $$u in $$the denominator, which can be integrated term-by-term. This step prepares the integral for tabular integration or direct integration of simpler terms. Set up the tabular integration by choosing parts: let the algebraic part (powers of $$u) be $$differentiated repeatedly until it becomes zero, and the remaining part (powers of $$u in $$the denominator) be integrated repeatedly. Construct the table with derivatives and integrals accordingly. Use the tabular integration method to combine the derivatives and integrals with alternating signs, then sum the resulting terms to write the integral in terms of $$u. $$Finally, substitute back $$u = x - 1 to $$express the answer in terms of $$x.$$

78. Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
e. ∫ (2x² - 3x) / (x - 1)³ dx

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Concetti chiave

Tabular Integration Method

Polynomial Division

Integration of Rational Functions