AP Calculus AB Exam Question:

Let f(x) be a function defined as follows:

f(x) = (x^2 + 3x - 2) / (x - 1) + (x + 5)e^x

(a) Find the antiderivative F(x) of f(x) with respect to x. (b) Evaluate the definite integral ∫[1, 2] f(x) dx.

Answer:

(a) To find the antiderivative F(x) of f(x), we will split the function into two separate integrals:

F(x) = ∫[(x^2 + 3x - 2) / (x - 1)] dx + ∫[(x + 5)e^x] dx

First, let's evaluate the integral ∫[(x^2 + 3x - 2) / (x - 1)] dx:

We can use long division or synthetic division to divide (x^2 + 3x - 2) by (x - 1):

        x + 4

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x - 1 | x^2 + 3x - 2 - (x^2 - x) ___________ 4x - 2 - (4x - 4) ___________ 2

Now, the integral becomes:

∫[(x^2 + 3x - 2) / (x - 1)] dx = ∫(x + 4) dx + ∫(2 / (x - 1)) dx

Using the power rule of integration, the first integral becomes:

∫(x + 4) dx = (1/2)x^2 + 4x + C1, where C1 is the constant of integration.

For the second integral, we can use the natural logarithm rule:

∫(2 / (x - 1)) dx = 2 ln |x - 1| + C2, where C2 is the constant of integration.

Now, let's evaluate the second part of the original function:

∫[(x + 5)e^x] dx:

Using integration by parts, we let u = (x + 5) and dv = e^x dx.

Differentiating u, we get du = dx, and integrating dv, we get v = e^x.

Now, the integration by parts formula states that:

∫ u dv = uv - ∫ v du

Applying this formula, we have:

∫[(x + 5)e^x] dx = (x + 5)e^x - ∫ e^x dx = (x + 5)e^x - e^x + C3, where C3 is the constant of integration.

Finally, the antiderivative F(x) of f(x) is:

F(x) = (1/2)x^2 + 4x + C1 + 2 ln |x - 1| + (x + 5)e^x - e^x + C3.

(b) To evaluate the definite integral ∫[1, 2] f(x) dx, we simply substitute the limits of integration into the antiderivative F(x) and evaluate:

∫[1, 2] f(x) dx = F(2) - F(1)

Substituting the limits, we have:

= [(1/2)(2)^2 + 4(2) + C1 + 2 ln |2 - 1| + (2 + 5)e^2 - e^2 + C3] - [(1/2)(1)^2 + 4(1) + C1 + 2 ln |1 - 1| + (1 + 5)e^1 - e^1 + C3]

Simplifying further, we have:

= 5/2 + 8 + C1 + 2 ln 1 + 7e^2 - e^2 - 5/2 - 4 - C1 - 2 ln 0 + 6e - e - C3

Since ln 0 is undefined, we need to consider the limit as x approaches 0 for ln |x - 1|. As x approaches 0, |x - 1| becomes -1, so ln |-1| = ln 1 = 0.

Therefore, the expression further simplifies to:

= 8 + 2(0) + 7e^2 - e^2 - 4 + 6e - e - C3 = 4 + 6e - e + 6e^2

Hence, the value of the definite integral ∫[1, 2] f(x) dx is 4 + 6e - e + 6e^2.