Provide an Appropriate Response $\int _ { 0 } ^ { \infty } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x$

question 369

Essay

Provide an appropriate response.
-i) Show that $\int _ { 0 } ^ { \infty } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x$ diverges, and hence $\int _ { - \infty } ^ { \infty } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x$ diverges.
ii) Show that $\lim _ { b \rightarrow \infty } \int _ { - b } ^ { b } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x = 0$ .

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Definitions:

Law-Of-Numbers

A principle suggesting that larger numbers of trials or samples lead to more stable and accurate averages and outcomes.

Group Think

A situation in collective decision-making processes where the pressure to agree or fit in causes outcomes that are irrational or lacking in effectiveness.

Cohesiveness

The degree to which members of a group are attracted to each other and motivated to stay in the group, often leading to improved performance and satisfaction.

Group Size

Refers to the number of individuals constituting a group, affecting dynamics such as cohesion, participation, and overall behavior within the group.