Question

Is there any shortcut methods for integration of logarthmic or exponential function ?

Answer 1

Kinda.

Explanation:

Really, it depends on your point of view. It's not really a shortcut as it's mostly just using their properties.

The defining aspects of logarithmic functions, with respect to differentiation is that if we have a function say:

`f(x) = ln(g(x))`

By the chain rule its derivate is

`f^'(x) = g^'(x)/g(x)`

So you can see it as a shortcut, if you have a logarithmic function your derivate will be "whatever's inside in the bottom and the derivate of that on the top". Which is useful as logarithms have some useful properties (and in fact, with this fact the quotient rule is even more obsolete)

As for exponential functions, if we have:

`f(x) = e^(g(x))`

By the chain rule its derivate is

`f^'(x) = g^'(x)e^(g(x))`

So you can see it as a shortcut, if you have an exponential function your derivate will be "just multiply the function by the derivative of whatever is being taken the exponential".

Remembering that

`a^x = e^(ln(a)*x)`

And that

`log_a(x) = ln(x)/ln(a)`