4. For some basic functions (like arctan z, Inz, and tanz) we can't find an antiderivative just by invert- ing a derivat

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4 For Some Basic Functions Like Arctan Z Inz And Tanz We Can T Find An Antiderivative Just By Invert Ing A Derivat 1 (98.81 KiB) Viewed 56 times

4. For some basic functions (like arctan z, Inz, and tanz) we can't find an antiderivative just by invert- ing a derivative formula we already know. These examples do have antiderivatives, however - just not "nice" ones. Have GeoGebra compute this indefinite integral, and write the answer: S arctan r dr = Notice that GeoGebra adds an arbitrary con- stant to the end of antiderivatives (like c₁, or C₂, etc.) and makes a slider for the con- stant. GeoGebra also sketches the graph of the antiderivative, and as you use the slider to change the constant, GeoGebra changes the graph. At right, sketch a graph of the an- tiderivatives of arctan x that go with the con- stants 0, 2 and -3. 5. The function f(x) = e-² is an important function in many applications of calculus, but it is also an example of a simple function whose antiderivative cannot be made from algebraic, trigonometric, exponential or logarithmic operations. (a) Have GeoGebra compute fe-² dx. Write down the answer you get: (b) A numerical approximation to an integral like fe bra compute it and round to four decimal places: dx is important in statistics. Have GeoGe-