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

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.