1. Find the following integrals.

(a) $\int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) dx$

(a) $\int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) dx$

(b) $\int (r^2 + r + 1) e^r \; dr$

2. Suppose that $f$ and $g$ are continuous functions and
$\int_1^5 f(x) dx = -1, \; \int_5^7 f(x) dx = 2, \quad \int_1^7 g(x) dx = 3$
Use the properties of definite integrals to find

(a) $\int_1^7 f(x) dx$

(a) $\int_1^7 f(x) dx$

(c) $\int_1^7 (3f(x) - 2g(x)) dx$

3. Find the following definite integrals.

(a) $\int_{-1}^{-\frac{1}{2}} t^{-2} \sin \left( 1 + \frac{1}{t} \right) dt$

(a) $\int_{-1}^{-\frac{1}{2}} t^{-2} \sin \left( 1 + \frac{1}{t} \right) dt$

(b) $\int_1^2 \frac{e^{2 \ln x}}{x} dx$ (Hint: simplify the integrand)

4. Use partial fractions to find the following integral.
$\int \frac{x^2 + 1}{(x^2 - 1)(x + 1)} dx$

5. Find the volume of the solid generated by revolving the region bounded by $y = 2 - x^2$ and the line $y = 1$ and $x = -1$ about the $x$-axis using disc/washer method.

6. Let $f(x) = kx^{\frac{3}{2}}$ be a function on the interval $[0, 1]$, where $k$ is a positive constant.

(a) Find $k$ for which $f(x)$ will be a probability density function of some random variable.

(a) Find $k$ for which $f(x)$ will be a probability density function of some random variable.

(b) Set up an integral for the expectation random variable. You don't need to evaluate it.

(c) Set up an integral for the variance of the random variable. You don't have to evaluate it.

7. Consider the integral $\int_{-1}^{2} \sin x \; dx$.

(a) Use Simpson's Rule with $n = 8$ to approximate the integral. You do not have to simplify the answer.

(a) Use Simpson's Rule with $n = 8$ to approximate the integral. You do not have to simplify the answer.

(b) If you wanted to use the trapezoid rule to approximate this integral with an error of less than $0.01$, how many trapezoids would you need? The error $E$ in approximating $\int_a^b f(x) dx$ using the trapezoid rule is
$|E| \le \frac{(b - a)^3}{12n^2} \left[ \max_{a \le x \le b} | f''(x) | \right]$

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