In integrals we were taught substitution, but we were not said an intuitive way of thinking about it.
In some questions we substitute x=tan(¢) or something like that but how can we know they r equal. They are both varying how can we say the rate of change of the LHS equals the RHS how can a linear function equals this whole mess of infinite series. I know a bit about Taylor series but don't know why do we substitute some wierd things in an integral. Can someone help me out.
The idea of a substitution is to make the resulting integral easier than the one before. A classic one for trigonometric substitution you mention is $$ \int \frac{dx}{1+x^2}, $$ which is not at all clear how to find the anti-derivative for. The substitution $x = \tan t$ makes it easier since $1 + x^2 = 1 + \tan^2 t = \sec^2 t$ and $dx = \sec^2 t dt$, so $$ \int \frac{dx}{1+x^2} = \int \frac{\sec^2 t\ dt}{1+\tan^2 t} = \int \frac{\sec^2 t\ dt}{\sec^2 t} = \int dt = t + C = \arctan x + C, $$ since the resulting integral became trivial.
The intuitive reason you can do this is that substitution for integrals is like the chain rule for derivatives. In other words, Chain Rule says $$ \frac{d}{dx} [f(g(x))] = f'(g(x)) g'(x), $$ or equivalently in integral form, $$ \int d[f(g(x)] = \int f'(g(x))g'(x)dx $$ and the analog of what we do is substitute $u = g(x), du = g'(x) dx$ into the RHS to get $$ \int d[f(g(x)] = \int f'(g(x))g'(x)dx = \int f'(u) du = \int d[f(u)]. $$
So as you can see, substitution is just an intuitive way of looking the implications of Chain Rule...