Why is the integral of dx 1 but the intgral of dx ^ n not possible

Question

myininaya · Answer

what

anonymous · Answer

$$\int\limits_{}^{}dx = 1 $$ but $$\int\limits_{}^{}dx ^{3} $$ does not = 1^3

anonymous · Answer

Why can dx not be integrated if it has an exponent higher than 1?

anonymous · Answer

dx is not a term, or a variable, or anything other than an identifier that we are trying to integrate something.  There is no function "dx"; when you see something like 
$$\int\limits_{}^{} dx$$
it really means 
$$\int\limits_{}^{} 1 dx$$
Thus you're integrating 1.

anonymous · Answer

right, so If I come to a dx^3 by u-substitution shouldnt it just be like integrating 1^3

anonymous · Answer

No, because dx itself is not a function that you can raise to a power.  If you integrate 
$$\int\limits_{}^{} 1^{3} dx$$
you are still integrating 1.  You are not integrating "dx"

anonymous · Answer

You shouldn't come to a $$dx^3$$by u-substitution in the first place.