Thread: How to integrate this weird function.

1. How to integrate this weird function.

Hello, how would you integrate this:

I was stumped when I saw this question, as I couldn't use a substituition to solve this.
If somebody knows how to integrate this, could they please show me how they go about doing so?

Thanks

2. Re: How to integrate this weird function.

Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case v), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

(But we can maybe do without the derivatives which are one.)

Spoiler:

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

3. Re: How to integrate this weird function.

Thanks Tom
Your ballooning method is ingeniously clear and pretty on the eyes too

Thanks, I get it now.