# Question A

(1) Find the indefinite integral $\int (x + 1) \ln x \; dx$

(2) Compute $\int_0^{\pi/2} \sin x \cos^2 x \; dx$

# Question B

(1) Find the indefinite integral $\int xe^{2x} \; dx$.

(2) Compute $\int_0^{3} \frac{x}{\sqrt{x + 1}} \; dx$

# Question C

(1) Obtain the partial fraction decomposition of $\frac{x + 3}{x(x^2 - 1)}$.

(2) Find the indefinite integral $\int \frac{x + 3}{x(x^2 - 1)}dx$

(3) Does the improper integral $\int_1^2 \frac{x + 3}{x(x^2 - 1)} dx$ converge?

# Question D

(1) Find the indefinite integral $\int xe^{-x^2} dx$.

(2) Determine if the improper integral $\int_{-\infty}^{\infty} xe^{-x^2} dx$ converges. If it converges determine its value.

# Question E

(1) Find the area of the region bounded by the curves $y = x^2 - 1$ and $y = 1 -x$.

(2) Let $R$ be the region bounded by the curve $y = x^2$ and the line $y = 1$. Compute the volume of the solid obtained by revolving $R$ about the $x$-axis.

# Question F

Suppose a continuous random variable has probability density function $f(x) = \frac{3}{4}(1 - x^2), -1 \le x \le 1$.
(1) Compute $P(0 \lt X \le 1)$.

(2) Compute the mean $E(X)$.

(3) Compute the variance $V(X)$.

(1) $\int_0^{\infty} \frac{1}{1 + 5e^x} dx$.
(2) $\int_0^{\infty} \sin x \; dx$