Question

Calculus= Integrate 5^x dx please break down how answer is 1/6x^6+C

Answer 1

`\int\limits {5^x} \, dx = (5^x)/(ln\ x) + c`

Explanation:

Note that the integral of
`5^x` is not
`(1)/(6)x^6 + c`

The solution is as follows:

Given

`5^x`

Required

Integrate

Represent the given expression using integral notation

`\int\limits {5^x} \, dx`

This question can't be solved directly;

We'll make use of exponential rules which states;

`\int\limits {a^x} \, dx = (a^x)/(ln\ x) + c`

By comparing
`\int\limits {5^x} \, dx` with
`\int\limits {a^x} \, dx`;

we can substitute 5 for a;

Hence, the expression
`\int\limits {a^x} \, dx = (a^x)/(ln\ x) + c` becomes

`\int\limits {5^x} \, dx = (5^x)/(ln\ x) + c`

-------------------------------------------------------------------------------------

However, the integral of
`x^5` is
`(1)/(6)x^6 + c`

This is shown below:

Given that
`x^5`

Applying power rule;

Power rule states that

`\int\limits{x^n}\ dx = (x^(n+1))/(n+1) + c`

In this case (
`x^5`), n = 5;

So,
`\int\limits{x^n}\ dx= (x^(n+1))/(n+1) + c`

becomes

`\int\limits{x^5}\ dx = (x^(5+1))/(5+1) + c`

`\int\limits{x^5}\ dx = (x^(6))/(6) + c`

`\int\limits{x^5}\ dx= (x^(6))/(6) + c`