(A) Express $\int _ { 0 } ^ { x } \frac { \sin \left( t ^ { 2 } \right) } { t ^ { 2 } } d t$

question 269

Essay

(a) Express $\int _ { 0 } ^ { x } \frac { \sin \left( t ^ { 2 } \right) } { t ^ { 2 } } d t$ as a Maclaurin series.(b) Evaluate $\int _ { 0 } ^ { 03 } \frac { \sin \left( x ^ { 2 } \right) } { x ^ { 2 } } d x$ as a series.

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Definitions:

Y-Intercept

The point at which a line or curve intersects the y-axis of a graph, representing the value of the dependent variable when the independent variable is zero.

Least Squares Regression

A statistical method used to determine the best-fitting line through a set of points by minimizing the sum of the squares of the vertical distances (residuals) of the points from the line.

Slope

In mathematics and statistics, the slope refers to the measure of the steepness or incline of a line, quantitively defined as the ratio of the vertical change to the horizontal change between two points on the line.

Y-Intercept

The point where a line crosses the Y-axis of a graph, representing the value of y when x is zero.