What is the integral of [(2x-y)^2]dx?
To find the integral of the expression (2x - y)^2 with respect to x, we can expand it first to simplify the integration.
(2x - y)^2 = (2x - y)(2x - y)
= 4x^2 - 2xy - 2xy + y^2
= 4x^2 - 4xy + y^2
Now that we have the simplified expression, we can integrate it term by term.
∫(4x^2 - 4xy + y^2) dx = ∫4x^2 dx - ∫4xy dx + ∫y^2 dx
To integrate each term, we use the power rule.
The integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1.
∫4x^2 dx = (4/3)x^3 + C1
∫4xy dx = 2xy^2 + C2
(Note that when integrating with respect to x, y is treated as a constant.)
∫ y^2 dx = y^2x + C3
Combining all the results, the integral of (2x - y)^2 with respect to x is:
(4/3)x^3 - 2xy^2 + y^2x + C, where C = C1 + C2 + C3.