Finding the integral of e^4x using u-substitution

Question

A question on my homework involves finding the integral of e^4x, and since the integral of e^x is just the same (e^x (+c)) I assumed that this would be the case for e^4x. However upon looking it up I see I am wrong, and that the actual answer is 1/4 e^4x. The only explanation I can find for this is through something called u-substitution, a concept I am unfamiliar with. Perhaps there is another way, but could anyone explain why this is the case? Is there a general rule?

Answer 1

Hi Havar,

u-substitution is definitely the best way to understand this. However, you can also just remember the derivative rule that given a real number r, we have:

(d/dx)(e^rx) = re^rx

which means that the integral (antiderivative) of e^rx is (1/r)e^rx (just divide both sides by r).

I hope that helps! I would also be happy to explain how u-substitution works if that would be useful.

Answer 2

In the integrand set u=4x, then du=4 dx i.e. dx=(1/4) du

The integrand then becomes (1/4) e^u du....and you take it from there.

Answer 3

use u(x)=4x in this case.

then du=4dx,,,dx=du/4

then have to find

∫e^udu/4

bring out 1/4

(1/4)∫e^udu=(1/4/)e^u+C=(1/4)e^4x+C

that is how to use u(x) for this problem

some students very much try to use u(x)

substitution as much as possible.

but often it is faster to evaluate the integral

without u(x)

see g(x)=∫e^4xdx

recognize a constant is missing from dx

d(e^4x)/dx=4e^4x,,,,need integrand to be e^4x*4dx

multiply by 1=(1/4)*4

now g(x)=(1/4)∫e^4x4dx=(1/4)e^4x+c