AP Calculus AB Exam Question:

Consider the function $f(x) = \frac{5}{x^2}$.

a) Find the indefinite integral $\int f(x) dx$.

b) Find the definite integral $\int_{1}^{2} f(x) dx$.

Answer:

a) To find the indefinite integral $\int f(x) dx$, we need to integrate the function $f(x) = \frac{5}{x^2}$ with respect to $x$. Applying the power rule of integration, we have:

Using the power rule of integration, we can rewrite $x^{-2}$ as $x^{-2} = \frac{1}{x^2}$.

Now, using the power rule of integration, we integrate $\frac{1}{x^2}$:

where $C$ is the constant of integration.

Therefore, $\int f(x) dx = - \frac{5}{x} + C$.

b) To find the definite integral $\int_{1}^{2} f(x) dx$, we use the fundamental theorem of calculus. From part a), we know that $\int f(x) dx = - \frac{5}{x} + C$. Now, we evaluate the definite integral:

Hence, $\int_{1}^{2} f(x) dx = \frac{5}{2}$.