Question

What is `f(x) = int (x-2)^3 dx` if `f(2) = 0`?

Answer 1

`f(x) = (x-2)^4/4`

Explanation:

Firstly we need to calculate the indefinite integral given by:

`f(x) = int \ (x-2)^3 \ dx`

There are a couple of approaches;

Method 1 - Substitution

We could use a simple change of variable, Let

`u = x-2 => (du)/dx = 1`

If we perform the substitution then we get:

`f(x) = int \ u^3 \ du`

This is a trivial integral and we can easily evaluate using the power rule to get:

`f(x) = u^4/4 + C`

Restoring the substituting we get:

`f(x) = (x-2)^4/4 + C`

Using the given condition `f(2)=0` we can evaluate the constant `C`:

`(2-2)^4/4 + C = 0 => C =0`

Hence, the solution is:

`f(x) = (x-2)^4/4`

Method 2 - Term By Term Integration

We could also evaluate the integral by expanding the binomial expression and integrating term by term:

`f(x) = int \ (x-2)^3 \ dx`
`" " = int \ (x)^3 + 3(x)^2(-2) + 3(x)(-2)^2 + (-2)^3 \ dx`
`" " = int \ x^3 - 6x^2 + 12x -8 \ dx`

Now integrating term by term we get:

`f(x) = x^4/4 - 2x^3 + 6x^2 - 8x + C`

Using the given condition `f(2)=0` we can evaluate the constant `C` (this is a diffret constant to that of Method 1):

`4 - 16 + 24 - 16 + C = 0 => C=4`

Hence, the solution is:

`f(x) = x^4/4 - 2x^3 + 6x^2 - 8x +4`

Which is the same as that from Method 1.