Consider the function $f ( x ) = \frac { 1 } { x ^ { 2 } - k x }$, where $k$ is a nonzero constant. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { k - 2 x } { \left( x ^ { 2 } - k x \right) ^ { 2 } }$.
(a) Let $k = 3$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 3 x }$. Write an equation for the line tangent to the graph of $f$ at the point whose $x$-coordinate is 4.
(b) Let $k = 4$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 4 x }$. Determine whether $f$ has a relative minimum, a relative maximum, or neither at $x = 2$. Justify your answer.
(c) Find the value of $k$ for which $f$ has a critical point at $x = -5$.
(d) Let $k = 6$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 6 x }$. Find the partial fraction decomposition for the function $f$. Find $\int f ( x ) \, dx$.

Let $f$ be the function defined by $f(x) = \frac{3}{2x^2 - 7x + 5}$.
(a) Find the slope of the line tangent to the graph of $f$ at $x = 3$.
(b) Find the $x$-coordinate of each critical point of $f$ in the interval $1 < x < 2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers.
(c) Using the identity that $\frac{3}{2x^2 - 7x + 5} = \frac{2}{2x - 5} - \frac{1}{x - 1}$, evaluate $\int_{5}^{\infty} f(x)\, dx$ or show that the integral diverges.
(d) Determine whether the series $\sum_{n=5}^{\infty} \frac{3}{2n^2 - 7n + 5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.

Consider the family of functions $f ( x ) = \frac { 1 } { x ^ { 2 } - 2 x + k }$, where $k$ is a constant.
(a) Find the value of $k$, for $k > 0$, such that the slope of the line tangent to the graph of $f$ at $x = 0$ equals 6.
(b) For $k = - 8$, find the value of $\int _ { 0 } ^ { 1 } f ( x ) \, dx$.
(c) For $k = 1$, find the value of $\int _ { 0 } ^ { 2 } f ( x ) \, dx$ or show that it diverges.

Recall that $x$ is a fixed element of $]0;1[$. Establish the identity
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + x } + \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + 1 - x }$$

Calculate the following indefinite integral: $$\int \frac{x^{2} + x + 2}{x^{3} - px^{2}} \, dx$$ where $p$ is a real constant.

$$\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } } { x ^ { 2 } - 1 } d x$$
What is the value of the integral?
A) $1 + \ln \left( \frac { 4 } { 3 } \right)$
B) $1 + \ln \left( \frac { 9 } { 2 } \right)$
C) $2 + \ln \left( \frac { 3 } { 2 } \right)$
D) $2 + \ln \left( \frac { 5 } { 3 } \right)$
E) $3 + \ln \left( \frac { 1 } { 3 } \right)$

$\int _ { 4 } ^ { 5 } \frac { x + 1 } { x ^ { 2 } - 5 x + 6 } d x$\ What is the value of the integral?\ A) $5 \ln 3 - \ln 2$\ B) $5 \ln 2 - 2 \ln 3$\ C) $3 \ln 2 + 2 \ln 3$\ D) $2 \ln 2 + 3 \ln 3$\ E) $7 \ln 2 - 3 \ln 3$