Question

Find the indefinite integral?

Find the indefinite integral without using the substitution method.

`int2^x(3^x+5^x)`

Answer 1

`int 2^x(3^x+5^x) dx = 6^x/ln 6+10^x/ln 10 + C`

Explanation:

Note that in general:

`d/(dx) a^x = d/(dx) e^(x ln a) = (ln a) e^(x ln a) = (ln a) a^x`

So:

`int a^x dx = a^x/(ln a) + C`

So we find:

`int 2^x(3^x+5^x) dx = int 6^x+10^x dx`

`color(white)(int 2^x(3^x+5^x) dx) = 6^x/ln 6+10^x/ln 10 + C`