find the antiderivative of f(x) = x^3(x-2)^2
Someone help me please!
rather than resorting to some formula method, I would expand the whole thing, and then integrate each of the simple terms.
Would I square (x-2) first and then multiply it by x cubed, or x cubed multiplied by (x-2) and then square the answer? Order of operations is parenthesis, exponents, multiplication and division (left to right), and then addition and subtraction ( left to right), isn't it?
you would square x-2 first, the squaring does not include the x^3, or else the whole thing would have been in brackets, should have had...
x^3(x^2 - 4x+4)
= x^5 - 4x^4 + 4x^2
now integrate that
Integrate? Meaning?
To find the antiderivative of the function f(x) = x^3(x-2)^2, we can use the method of integration by parts. This method involves using the product rule in reverse.
Step 1: Recall the product rule:
If u(x) and v(x) are differentiable functions of x, then the derivative of their product, u(x)v(x), with respect to x is given by:
(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x).
Step 2: Apply the integration by parts formula:
∫u(x)v'(x) dx = u(x)v(x) - ∫v(x)u'(x) dx.
In the case of our function f(x) = x^3(x-2)^2, we can choose u(x) = x^3 and v'(x) = (x-2)^2.
Step 3: Find u'(x) and v(x):
To find u'(x), take the derivative of u(x) = x^3 with respect to x:
u'(x) = 3x^2.
To find v(x), integrate v'(x) = (x-2)^2 with respect to x:
We can expand (x-2)^2 to get (x^2 - 4x + 4). Then we integrate each term separately:
∫v'(x) dx = ∫(x^2 - 4x + 4) dx = (1/3)x^3 - 2x^2 + 4x + C.
Step 4: Apply the formula:
Now we apply the integration by parts formula:
∫x^3(x-2)^2 dx = u(x)v(x) - ∫v(x)u'(x) dx.
= x^3[(1/3)x^3 - 2x^2 + 4x + C] - ∫[(1/3)x^3 - 2x^2 + 4x + C] (3x^2) dx.
Simplify the expression:
= (1/3)x^6 - 2x^5 + 4x^4 + Cx^3 - [(1/3)x^6 - 2x^5 + 4x^4 + Cx^3] - ∫(1/3)x^6 - 2x^5 + 4x^4 + Cx^3 dx.
Finally, we can integrate the remaining terms and combine like terms.
Keep in mind that the constant C represents the constant of integration, which is added whenever we integrate.
Thus, the antiderivative of f(x) = x^3(x-2)^2 is:
(1/3)x^6 - 2x^5 + 4x^4 + Cx^3.