A smart way to avoid a lot of Inverse hassle?

Hi, Calculus is about logic & it's great because

if you figure out the underlay of specific things

you can sidestep memorizing a heckload of

information.

To me the only hard thing is finding these *tricks*,

I'm specifically having a problem with integrating

inverse trigonometric functions.

For taking the derivative it's simple,

$\displaystyle y = sin^{-1}(2x^2)$

$\displaystyle siny = 2x^2$

$\displaystyle cosy \cdot \frac{dy}{dx} = 4x$

$\displaystyle \frac{dy}{dx} = \frac{4x}{cosy}$

$\displaystyle \frac{dy}{dx} = \frac{4x}{\sqrt{1 - sin^2y} }$

$\displaystyle \frac{dy}{dx} = \frac{4x}{\sqrt{1 - 4x^4} }$

and voila!

No memorizing whatever it is that litters up

the engineering book I foolishly bought.

However, for integrating these functions I don't see

the easy process.

Like, there are 6 functions all differing very slightly

and there's no way I'll be able to memorise them.

Every source I've checked tells me that because I know

the derivatives I can just reverse the process & copy

the form & write the integral. Yay!

Except, I'd have to go through up to a possible 6 side

calculations of derivatives to search out which

form my current integral is in so I can just magially

copy it.

Is there no better way than rote memorization?

Is there no simple logical model to follow that

generalizes to all 6 functions like taking derivatives

of them all has?