Question

Integrate the following. ∫
`5x^(4) dx`

Option A:
`x^(5)`
Option B:
`5x^(5) +C`
Option C:
`5x^(5)`
Option D:
`x^(5) + C`

Answer 1

The answer is D I just had this on a test.

Answer 2

`\displaystyle D) {x}^(5) + \rm C`

Explanation:

we would like to integrate the following Integral:

`\displaystyle \int 5 {x}^(4) \, dx`

well, to get the constant we can consider the following Integration rule:

`\displaystyle \int c{x} ^(n) \, dx = c\int {x}^(n) \, dx`

therefore,

`\displaystyle 5\int {x}^(4) \, dx`

recall exponent integration rule:

`\displaystyle \int {x} ^(n) \, dx = \frac{ {x}^(n + 1) }{n + 1}`

so let,

• 
  `n = 4`

Thus integrate:

`\displaystyle = 5\left( \frac{ {x}^(4+ 1) }{4 + 1} \right)`

simplify addition:

`\displaystyle = 5\left( \frac{ {x}^(5) }{5} \right)`

reduce fraction:

`\displaystyle = {x}^(5)`

finally we of course have to add the constant of integration:

`\displaystyle \boxed{ {x}^(5) + \rm C}`

hence,

our answer is D)