Exam 8: Basic Integration Rules

$\text { Evaluate the limit } \lim _ { x \rightarrow 8 } \frac { - 2 x ^ { 2 } + 25 - 72 } { x - 8 } \text { first by using techniques from Chapter } 1$ then by using L'Hopital's Rule.

$\text { Find } \int e ^ { 2 x } \sqrt { 1 - e ^ { 4 x } } d x$

Find the indefinite integral. $\int \sec ^ { 6 } 6 x d x$

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 10 } { x ^ { 2 } + 10 x }$

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \frac { 3 \left( e ^ { x } - 1 - x \right) } { 10 x ^ { 3 } } \text { using L'Hopital's Rule if necessary. }$

Suppose a model for the ability M of a child to memorize, measured on a scale from 0 to 10 , is given by $M = 1 + 1.4 t \ln t , 0 < t \leq 4$ where $t$ is the child's age in years. Find the average value of this model between the child's first and second birthdays. Round your answer to three decimal places.

Find the indefinite integral. $\int \left[ r + \frac { 5 } { ( r - 2 ) ^ { 4 } } \right] d r$

$\text { Use partial fractions to find } \int \frac { x ^ { 2 } + x + 5 } { x ^ { 4 } + 10 x ^ { 2 } + 25 } d x \text {. }$

$\text { Use substitution to find the integral } \int \frac { e ^ { x } } { \left( e ^ { 2 x } + 1 \right) \left( e ^ { x } - 3 \right) } d x \text {. }$

Find the definite integral. $\int _ { 5 } ^ { 6 } \frac { \sqrt { x ^ { 2 } - 25 } } { x ^ { 2 } } d x$

$\text { Find the area between the } x \text {-axis and the graph of the function } y = \frac { 2 } { x ^ { 2 } + 1 } \text {. }$

Find the definite integral. $\int _ { 0 } ^ { 3 } x ^ { 2 } e ^ { - x ^ { 3 } } d x$

Find the indefinite integral. $\int \sec ^ { 5 } 2 x \tan 2 x d x$

Find the definite integral. $\int _ { 0 } ^ { 12 } \frac { 8 x } { \sqrt { x ^ { 2 } + 25 } } d x$

$\text { Find the indefinite integral by making the substitution } x = 9 \sin \theta \text {. }$ $\int \frac { 9 } { x ^ { 2 } \sqrt { 81 - x ^ { 2 } } } d x$

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 } \frac { \arcsin ( 13 x ) } { 2 x } \text { using L'Hopital's Rule if necessary. }$

Evaluate the definite integral $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } - x } { x ^ { 2 } + x + 1 } d x$

$\text { Determine whether the improper integral } \int _ { 9 } ^ { 11 } \frac { 2 } { ( x - 10 ) ^ { 2 } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.