To find the possible values of a, we need to use the fundamental theorem of calculus, which states that if f is a continuous function on the interval [a, b], then
∫ (a, b) f(t) dt = F(b) - F(a),
where F is any antiderivative of f.
In this case, we are given that
∫ (a, x) f(t) dt = 3x^2 + 12x.
Differentiating both sides with respect to x, we get
f(x) = d/dx (3x^2 + 12x) = 6x + 12.
To find the antiderivative of f(x), we integrate the above equation:
∫ f(x) dx = ∫ (6x + 12) dx = 3x^2 + 12x + C,
where C is a constant of integration.
Therefore, we have
F(x) = 3x^2 + 12x + C.
Now, using the fundamental theorem of calculus, we have
∫ (a, x) f(t) dt = F(x) - F(a) = (3x^2 + 12x + C) - (3a^2 + 12a + C) = 3x^2 - 3a^2 + 12x - 12a.
Comparing this with the given expression
∫ (a, x) f(t) dt = 3x^2 + 12x,
we get
3x^2 - 3a^2 + 12x - 12a = 3x^2 + 12x.
Simplifying this equation, we get
-3a^2 - 12a = 0.
Dividing both sides by -3, we get
a^2 + 4a = 0.
Factoring out a, we get
a(a + 4) = 0.
Therefore, the possible values of a are a = 0 and a = -4.
To verify that these values of a are correct, we can check that the given expression holds for each of these values.
If a = 0, then
∫ (0, x) f(t) dt = 3x^2 + 12x,
which is the given expression.
If a = -4, then
∫ (-4, x) f(t) dt = 3x^2 + 12x - (3(-4)^2 + 12(-4)) = 3x^2 + 12x + 48 - 48 = 3x^2 + 12x,
which is again the given expression.
Therefore, the possible values of a are a = 0 and a = -4, and these values are verified to be correct.