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