Need an explanation please - integrals.

Question

anonymous · Answer

I am just wondering how 5^2x goes into 5^2x/2ln5 .. i know the formula, i'm just wondering where the 2 comes from ... is it because of the 5^2x, so since it's 2x i have to take 1/2 outside the integral sign?

also... i am wondering how to get the integral of 5^2x/2ln5x. i know what the answer is, i just don't know how to get it.

anonymous · Answer

$$
{d\over dx}(a^x)=a^x\ln(a)\
\implies \int a^xdx=\frac{a^x}{\ln a}
$$
now, set 
$$
u=2x\implies dx=\frac{du}{2}\
\int5^{2x}dx=\int5^u\frac{du}{2}=\frac{5^u}{2\ln5}+C\
=\frac{5^{2x}}{2\ln5}+C
$$

anonymous · Answer

okay now i'm just wondering how i would integrate that answer. (im doing integration by parts)

anonymous · Answer

that is the answer. no need to use integration by parts.

amistre64 · Answer

5^(2x) = 25^x

anonymous · Answer

ok

experimentx · Answer

just old substitution with this trick ... |dw:1363623298064:dw|

anonymous · Answer

can you please explain a little more?

experimentx · Answer

|dw:1363623465329:dw|