Integration by substitution is a tricky thing to master, I'm learning it from books and at some place my book has done something like this
$$\int \csc\theta \cot\theta \; d\theta = ?$$ $$ \int \frac{\cos\theta}{\sin^2\theta} \;d\theta$$ Let $u = \sin\theta$, $du = \cos\theta \;d\theta$ $$\int u^{-2} \;du$$ $$ -1/u = -1/\sin\theta = -\csc\theta$$
But it was very apparent that the integral was one of the standard one and using the substitution rule was redundant (it may that book wanted to show me the power of substitution rule).
I need some personal notes or something which you have made for your lectures or some personal real-life discovery about using the substitution rule in integration. I know this is an easy thing to master if some experienced man helps me.
Thank you.