JEE MAIN - Mathematics (2025 - 7th April Evening Shift - No. 23)
If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\alpha+\beta+\gamma$ is equal to ___________.
Answer
19
Explanation
$$\begin{aligned}
& \int\left(\frac{1}{x^2}+\frac{1}{x^4}\right)\left(\frac{3}{x}+\frac{1}{x^3}\right)^{\frac{1}{23}} d x \\
& \text { using } t=\frac{3}{x}+\frac{1}{x^3} \Rightarrow d t=-3\left(\frac{1}{x^2}+\frac{1}{x^4}\right) d x \\
& \int \frac{t^{1 / 23} d t}{-3}=\frac{t^{24 / 23}}{\left(\frac{24}{23}\right)(-3)}+C \\
& \Rightarrow \alpha=23 \beta=-1 \gamma=-3 \\
& \alpha+\beta+\gamma=19
\end{aligned}$$
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