JEE PYQ: Indefinite Integration Question 1
Question 1 - 2021 (16 Mar Shift 2)
For real numbers $\alpha, \beta, \gamma$ and $\delta$, if
$$\int \frac{(x^2 - 1) + \tan^{-1}\left(\frac{x^2+1}{x}\right)}{(x^4 + 3x^2 + 1)\tan^{-1}\left(\frac{x^2+1}{x}\right)},dx = \alpha\log_e\left(\tan^{-1}\left(\frac{x^2+1}{x}\right)\right) + \beta\tan^{-1}\left(\frac{\gamma(x^2-1)}{x}\right) + \delta\tan^{-1}\left(\frac{x^2+1}{x}\right) + C$$
where C is an arbitrary constant, then the value of $10(\alpha + \beta\gamma + \delta)$ is equal to ______.
Show Answer
Answer: 6
Solution
Put $\tan^{-1}\left(x + \frac{1}{x}\right) = t$. Split the integral into two parts. After substitution and simplification: $\alpha = 1$, $\beta = \frac{1}{2\sqrt{5}}$, $\gamma = \frac{1}{\sqrt{5}}$, $\delta = -\frac{1}{2}$. So $10(\alpha + \beta\gamma + \delta) = 10\left(1 + \frac{1}{10} - \frac{1}{2}\right) = 10 \cdot \frac{6}{10} = 6$.
BETA
