Find the indefinite integral: (2^-2x)dx
I used two different methods and got two different answers, so I'm wondering which method is correct..
1) integral of a^u = (1/lna)(a^u) + C
plugged it in and got (1/ln2)(2^-2x) +C
2) u substitution; u=-2x, du=-2dx
= (-1/2)(2^u) = -(2^-2x)/(2ln2) +C

Question

Find the indefinite integral: (2^-2x)dx
I used two different methods and got two different answers, so I'm wondering which method is correct..
1) integral of a^u = (1/lna)(a^u) + C
    plugged it in and got (1/ln2)(2^-2x) +C
2) u substitution; u=-2x, du=-2dx
   = (-1/2)(2^u) = -(2^-2x)/(2ln2) +C

zepdrix · Answer

Looks like method 2 worked out correctly :o
Notice that method 2 is actually a `u-sub` and then you simply did step 1 after that.

zepdrix · Answer

But I would encourage you to try to get comfortable `skipping` the u-substitution on problems like this.

If it's too difficult, then fineeee.
But you'll get it eventually.

zepdrix · Answer

Example:$$\Large\bf\sf (e^{2x})'\quad=\quad 2 e^{2x}$$
When you differentiate, the coefficient on x tells you to `multiply` by an extra 2 because of the chain rule.
$$\Large\bf\sf \int\limits e^{2x}\;dx\quad=\quad \frac{e^{2x}}{2}$$
When you integrate, the coefficient on x tells you to `divide` by an extra 2 to compensate for the chain rule would normally produce from the other direction.

zepdrix · Answer

So in your problem, (if you feel more comfortable doing a u-sub, then fineeeee)
you end up dividing by -2 because of your exponent.

anonymous · Answer

Ohhhhhhhhhhhhhhhh okay that makes WAY more sense, thanks! @zepdrix

anonymous · Answer

Does that apply to all integral rules, including the trig ones?

anonymous · Answer

@ganeshie8

ganeshie8 · Answer

when u skip u-substitution, u just need to make sure that the thing u cooked up differentiates back to the integrand.

skipping u-sub is called "advanced guessing"
you need to get use to advanced guessing... for simpler integrals like these

ganeshie8 · Answer

$\Large\bf\sf \int\limits e^{2x}\;dx\quad=\quad \frac{e^{2x}}{2} $

ganeshie8 · Answer

differentiate the "right side thing",
u should get back the integrand u started wid.

ganeshie8 · Answer

its a good idea to check/confirm when u do "advanced guessing"

ganeshie8 · Answer

btw, to answer ur original question, it doesnt work for everything. 
it works only in special cases like simple exponential functions  :
$\large    \int   a^{cx} dx$

simple trig functions :
$\large  \int \cos (ax + b) dx$

etc...

ganeshie8 · Answer

anyways it comes wid practice... we dont have to delve too much on this in the beginning  :)

anonymous · Answer

Okay, thanks so much :) @ganeshie8

ganeshie8 · Answer

np :)