Q.Choose the correct option from the given alternatives: $\int_2^3 \dfrac{dx}{x(x^3-1)} =$ (A) $\frac13\log\frac{208}{189}$ (B) $\frac13\log\frac{189}{208}$ (C) $\log\frac{208}{189}$ (D) $\log\frac{189}{208}$

Most definite integrals in this chapter are evaluated by first finding an indefinite integral (antiderivative) using a suitable technique — algebraic substitution, trigonometric substitution (such as $x=a\sin\theta$ for $\sqrt{a^2-x^2}$-type integrands, or the Weierstrass substitution $t=\tan(x/2)$ for rational functions of $\sin x$ and $\cos x$), rationalising a surd denominator by multiplying by its conjugate, splitting a rational function by partial fractions, using standard trigonometric product-to-sum or power-reduction identities, or integration by parts — and then applying the Fundamental Theorem of Calculus (evaluate at the upper limit, subtract the value at the lower limit). The one extra care needed for a definite integral (as opposed to an indefinite one) is that when the integration variable is substituted for a new variable, the limits of integration must also be converted to the new variable's corresponding values, so the final evaluation is done entirely in the new variable without ever having to substitute back. This is the default technique tried whenever none of the eight properties (Properties I–VIII) offers an obvious shortcut.