Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 20/refcontrol
- Exercises
Exercise Create referencenumber
Show by induction over , using integration by parts, that
holds.
The following exercises are about determining primitive functions. This includes the choice of suitable domains for the functions.
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise Create referencenumber
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 Create referencenumber
Let . Determine an antiderivative of the function
using the antiderivative of and the theorem about integration of inverse function.
Exercise Create referencenumber
Determine an antiderivative of the natural logarithm function using the antiderivative of its inverse function.
Exercise Create referencenumber
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 Create referencenumber
Compute the definite integralMDLD/definite integral
Exercise Create referencenumber
Compute the definite integral of the function
on .
Exercise Create referencenumber
Compute by an appropriate substitution an antiderivative of
Exercise Create referencenumber
Prove the relationship
for , only using rules for integration.
Exercise Create referencenumber
Determine the areas of the regions, surrounded by the blue curves, sketched on the right.
- Hand-in-exercises
Exercise (3 marks) Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise (2 marks) Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise (3 marks) Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Exercise (4 marks) Create referencenumber
Determine an antiderivativeMDLD/antiderivative (R) (primitive function) of the functionMDLD/function
Hint: Write the numerator polynomial using the denominator polynomial.
Exercise (4 marks) Create referencenumber
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
Exercise (5 marks) Create referencenumber
Let
be a differentiable functionMDLD/differentiable function (R) with for all . For what points does the area of the hatched surface have a local extremum?MDLD/local extremum Is it a minimum or a maximum?
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