First slide

Methods of integration

Question

$If \int \frac{(2 x - 1) dx}{x^{4} - 2 x^{3} + x + 1} = {Atan}^{- 1} [\frac{f (x)}{\sqrt{3}}] + C$ then

Moderate

A

$A = \frac{1}{\sqrt{3}}$
B

A=2
C

$f (x) = 2 x^{2} - 2 x - 1$
D

$f (x) = 2 x^{2} + 2 x + 1$

Solution

$\begin{array}{l} I = \int \frac{(2 x - 1) dx}{{(x^{2} - x)}^{2} - (x^{2} - x) + 1} \\ Put x^{2} - x = t \\ I = \int \frac{dt}{t^{2} - t + 1} = \int \frac{dt}{{(t - \frac{1}{2})}^{2} + \frac{3}{4}} \\ I = \int \frac{dt}{{(t - \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} \Rightarrow I = \frac{2}{\sqrt{3}} \tan^{- 1} (\frac{2 x^{2} - 2 x - 1}{\sqrt{3}}) + C \end{array}$

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