Evaluate the following integrals:
Question:

Evaluate the following integrals:

$\int\left\{\tan (\log x)+\sec ^{2}(\log x)\right\} d x$

Solution:

Let $I=\int\left[\tan (\log x)+\sec ^{2}(\log x)\right] d x$

$\log x=z \Rightarrow x=e^{z} \Rightarrow d x=e^{z} d z$

$I=\int\left(\tan z+\sec ^{2} z\right) e^{z} d z$

$\mathrm{f}(\mathrm{z})=\tan \mathrm{z} ; \mathrm{f}^{\prime}(\mathrm{z})=\sec ^{2} \mathrm{z}$

We know that, $\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\}=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{c}$

$I=x \tan (\log x)+c$

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