Question Number 190937 by Spillover last updated on 14/Apr/23
$$\mathrm{Show}\:\:\mathrm{that}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sech}\:\sqrt{\mathrm{x}}\:\mathrm{tanh}\:\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}}}=−\frac{\mathrm{2}}{\mathrm{cosh}\:\sqrt{\mathrm{x}}} \\ $$
Answered by ARUNG_Brandon_MBU last updated on 15/Apr/23
$${I}=\int\frac{\left(\mathrm{sech}\sqrt{{x}}\right)\left(\mathrm{tanh}\sqrt{{x}}\right)}{\:\sqrt{{x}}}{dx}=\int\frac{\mathrm{sinh}\sqrt{{x}}}{\:\sqrt{{x}}\mathrm{cosh}^{\mathrm{2}} \sqrt{{x}}}{dx} \\ $$$${t}=\mathrm{cosh}\sqrt{{x}}\:\Rightarrow{dt}=\frac{\mathrm{sinh}\sqrt{{x}}}{\mathrm{2}\sqrt{{x}}}{dx} \\ $$$${I}=\int\frac{\mathrm{2}}{{t}^{\mathrm{2}} }{dt}=−\frac{\mathrm{2}}{{t}}+{C}=−\frac{\mathrm{2}}{\mathrm{cosh}\sqrt{{x}}}+{C} \\ $$
Commented by Spillover last updated on 15/Apr/23
$$\mathrm{great} \\ $$