How to prove the inequality？ [closed]

Prove $$\int_{n+2 t\sqrt{n}}^{+\infty} \frac{\left(2 t^{2}+x\right)^{\frac{n}{2}-1} e^{-\frac{x}{2}}}{2^{\frac{n}{2}} \Gamma\left(\frac{n}{2}\right)} d x \leq 1$$ When $$t \geq 0$$ and $$n \in \mathbb{N^+}$$

closed as off-topic by StubbornAtom, RRL, Cesareo, Arnaud D., user1729May 10 at 12:26

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• What have you attempted? See math.meta.stackexchange.com/questions/9959/… – Jerry Chang May 10 at 1:43
• A good tag would have been gamma-function – Claude Leibovici May 10 at 4:33
• If $X \sim \text{Gamma}(n/2, 1/2)$, then the integral computes $E[(1+2t^2/x)^{n/2-1} \mathbf{1}_{X \ge n+2t\sqrt{n}}]$. But I am not sure if this observation is helpful. – angryavian May 10 at 6:21