`int (x^4 + 5^x) dx` Find the indefinite integral

For the given integral:` int (x^4+ 5^x) dx` , we may apply the basic integration property:

`int (u+v) dx = int (u) dx + int (v) dx` .

We can integrate each term separately.

`int (x^4+ 5^x) dx =int (x^4) dx + int (5^x) dx`

For the integration of first term:  `int (x^4) dx` ,

we apply the Power Rule for integration:

`int (x^n) dx = x^(n+1)/ (n+1) +C` .

Then,

`int (x^4) dx = x^(4+1)/(4+1) +C`

`int (x^4) dx = x^(5)/(5) +C`

For the integration of first term: `int (5^x) dx` , we apply the basic integration formula for exponential function :

`int (a^x) dx = a^x/ln(a) +C`  where  ` a!=1`

Then,

`int (5^x) dx =5^x/ln(5) +C`

 Combining the two integrations for the final answer:

 ` int (x^4+ 5^x) dx =x^(5)/(5) +x^(5)/(5) +C`