Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 74

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