Choose the correct quantity to fill in the blank:
Integral_______ dx = (ax2 - a2)4 + C.
to do this, we get the derivative of (ax^2 - a^2)^4 + c
assuming a is constant, the derivative is
4[(ax^2 - a^2)^3](2ax)
hope this helps~ :)
To determine the correct quantity to fill in the blank in the integral expression, let's start by looking at the given expression:
∫ _______ dx = (ax^2 - a^2)4 + C
We can apply the power rule of integration, which states that the integral of x^n is equal to (1/(n+1)) * x^(n+1), where n is any real number except -1.
In the given expression, the integral of the blank quantity will have to satisfy the power rule and result in (ax^2 - a^2)4 + C when integrated.
To find the correct quantity to fill in the blank, we'll apply the power rule and find the derivative of (ax^2 - a^2)4 + C with respect to x, which should yield the quantity in the blank:
First, let's apply the power rule to the expression (ax^2 - a^2)4:
Taking the derivative of (ax^2 - a^2)4 with respect to x, we get:
4(ax^2 - a^2)^3 * (2ax) = 8a(ax^2 - a^2)^3 * x.
Therefore, the correct quantity to fill in the blank is 8a(ax^2 - a^2)^3 * x.
Thus, the integral expression becomes:
∫ 8a(ax^2 - a^2)^3 * x dx = (ax^2 - a^2)4 + C.