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Question Number 190937 by Spillover last updated on 14/Apr/23

$$\mathrm{Show}\mathrm{that}$$$$\int \frac{\mathrm{sech}\sqrt{\mathrm{x}}\tanh \sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}}=-\frac{2}{\cosh \sqrt{\mathrm{x}}}$$

Answered by ARUNG_Brandon_MBU last updated on 15/Apr/23

$$I=\int \frac{\left(\mathrm{sech}\sqrt{x}\right)\left(\tanh \sqrt{x}\right)}{\sqrt{x}}dx=\int \frac{\sinh \sqrt{x}}{\sqrt{x}{\cosh }^2\sqrt{x}}dx$$$$t=\cosh \sqrt{x}\Rightarrow dt=\frac{\sinh \sqrt{x}}{2\sqrt{x}}dx$$$$I=\int \frac{2}{t^2}dt=-\frac{2}{t}+C=-\frac{2}{\cosh \sqrt{x}}+C$$

Commented by Spillover last updated on 15/Apr/23

$$\mathrm{great}$$

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