JEE PYQ: Indefinite Integration Question 23
Question 23 - 2019 (10 Apr Shift 1)
If $\int \frac{dx}{(x^2 - 2x + 10)^2} = A\left(\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right) + C$ where $C$ is a constant of integration, then:
(1) $A = \frac{1}{54}$ and $f(x) = 3(x-1)$
(2) $A = \frac{1}{81}$ and $f(x) = 3(x-1)$
(3) $A = \frac{1}{27}$ and $f(x) = 9(x-1)$
(4) $A = \frac{1}{54}$ and $f(x) = 9(x-1)^2$
Show Answer
Answer: (1)
Solution
Let $x - 1 = 3\tan\theta$. Then $(x-1)^2 + 9 = 9\sec^2\theta$ and $dx = 3\sec^2\theta,d\theta$. $I = \frac{1}{27}\int \cos^2\theta,d\theta = \frac{1}{54}(\theta + \frac{\sin 2\theta}{2}) + C = \frac{1}{54}\left(\tan^{-1}\frac{x-1}{3} + \frac{3(x-1)}{x^2-2x+10}\right) + C$. So $A = \frac{1}{54}$, $f(x) = 3(x-1)$.
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