the integral repressentation of the bessel function $j_\alpha(\lambda x)$
I have trouble to prove that $$|j_\alpha(\lambda x)|\leq e^{|Im(\lambda)||x|}$$ My attempt:\
$$|j_\alpha(\lambda x)|= \frac{2\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha-1/2)}\left|\int_{0}^1 (1-t^2)^{\alpha -1/2} cos(\lambda xt)dt \right|$$ $$\leq \frac{2\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha-1/2)}\int_{0}^1 \left|(1-t^2)^{\alpha -1/2} cos(\lambda xt)\right|dt$$ $$\leq \frac{2\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha-1/2)}\int_{0}^1 \left|(1-t^2)^{\alpha -1/2}\right|dt$$ now we let $t=cos \theta$. $$|j_\alpha(\lambda x)| \leq \frac{2\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha-1/2)}\int_{0}^{\pi/2} \left|(sin^2(\theta)^{\alpha -1/2}\right|d\theta$$.