Question

\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]

Answer 1

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

\[\text{ Put  log x = t }\]

\[ \Rightarrow x = e^t \]

\[ \Rightarrow dx = e^t dt\]

\[ \text{ ∴  I }= \int\left( \tan t + \sec^2 t \right) e^t dt\]

\[\text{ Here,} f(t) = \tan t\]

\[ \Rightarrow f'(t) = \sec^2 t\]

` \text{ let e}^t \tan(t) = p  `

\[\text{ Diff  both   sides  w . r . t t }\]

\[ e^t \left[ \tan t + \sec^2 t \right] = \frac{dp}{dt}\]

\[ \Rightarrow e^t \left[ \tan t + \sec^2 t \right]dt = dp\]

\[ ∴  I = \int dp\]

\[ = p + C\]

\[ = e^t \tan t + C\]

\[ = x \text{ tan (log x) }+ C\]