Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 25/en

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Warm-up-exercises

Exercise

Compute the definite integral


In the following exercises, which involve the determination of antiderivative functions, consider an appropriate domain of definition.

Exercise

Determine an antiderivative of the function


Exercise

Determine an antiderivative of the function


Exercise

Determine an antiderivative of the function


Exercise

Determine an antiderivative of the function


Exercise

Determine an antiderivative of the function


Exercise

Determine for which the function

has a maximum or a minimum.


Exercise

According to recent studies the student's attention skills during the day are described by the following function

Here is the time in hours and is the attention measured in micro-credit points per second. When should one start a one and a half hour lecture, such that the total attention skills are optimal? How many micro-credit points will be added during this lecture?


Exercise

Let be a real interval and let

be a continuous function with antiderivative . Let be an antiderivative of and let . Determine an antiderivative of the function


Exercise

Let . Determine an antiderivative of the function

using the antiderivative of and Theorem 25.4.


Exercise

Determine an antiderivative of the natural logarithm function using the antiderivative of its inverse function.


Exercise

Let

be a bijective, continuous differentiable function. Prove the formula for the antiderivative of the inverse function by the integral

using the substitution and then integration by parts.


Exercise

Compute by an appropriate substitution an antiderivative of


Exercise

Compute the definite integral of the function

on .


Exercise

Compute the definite integral of the function

on .




Hand-in-exercises

Exercise (4 points)

Compute the definite integral , where the function is


Exercise (3 points)

Determine an antiderivative of the function


Exercise (2 points)

Determine an antiderivative of the function


Exercise (4 points)

Determine an antiderivative of the function


Exercise (5 points)

Determine an antiderivative of the function


Exercise (5 points)

Let be a real interval and let

be a continuous function with antiderivative . Let be an antiderivative of and an antiderivative of . Let . Determine an antiderivative of the function




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