Question

Int (x/a+x^2)^3 dx = ?

Answer 1

`(1/7)x^7 + (1/(2a))x^6 + (3/(5a^2))x^5 + (1/(4a^3))x^4 + C`

Explanation:

This integral can be solved by expanding it.

Note that the following is true:

`(x/a + x^2)^3 = (x/a+x^2)(x/a + x^2)(x/a + x^2)`
`= (x^2/(a^2) + 2 x^3/a + x^4)(x/a + x^2)`
`= x^3/(a^3) + 2x^4/(a^2) + x^5/a + x^4/(a^2) + 2x^5/a + x^6`
`= x^6 + 3x^5/a + 3x^4 / (a^2) + x^3/(a^3)`

Making the above substitution, our integral is now:

`int ( x^6 + (3/a)x^5 + (3/(a^2))x^4 + (1/(a^3))x^3)dx`
`= (1/7)x^7 + (1/(2a))x^6 + (3/(5a^2))x^5 + (1/(4a^3))x^4 + C`

If it pleases you, you can find common denominators to make this a single term, but this form looks fine to me.