Question: evaluate the indefinite integral by using substitution
The answer key has
I tried the substitutions
1. doesn't really work so I think 2. is a better bet. I think I need to somehow get but I don't know how.
Also, just in case a picture helps...
... where (as usual) ...
represents the chain rule for differentiation - straight continuous lines differentiating down (integrating up) with respect to x, the straight dashed line similarly but with respect to the dashed balloon expression.
PS: I should have emphasised (and after Plato's comment below now can't resist)... read the diagram downwards to differentiate...
and upwards to integrate...
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
I think that the method of u-substitution is most deleterious, non-productive idea that matheducators have foisted off on poor students honestly trying to learn to do anti-differentiation. It gives a false sense of understanding of the actual process. Now I do agree with the philosopher of mathematics Michael Resnik (I admit being a friend), he is known as a structurelist: mathematics is the science of pattern.
To find an anti-derivative, one needs to look at the pattern of derivatives.