Evaluate: `int{tan(logx)+sec^2(logx)}\ dx`

Explanation

Let, `I=int{tan (log x)+sec ^{2}(log x)} d x`
` log x=z`
` Rightarrow x=e^z`
` Rightarrow d x=e^z d z`
` I=int(tan z+sec ^{2} z) e^{z} d z`
` f(z)=tan z ; f^{prime}(z)=sec ^{2} z `
We know that,
` int e^{x}{f(x)+f^{prime}(x)}=e^{x} f(x)+c I=x tan (log x)+c`