Methods in Calculus (Edexcel A Level Further Maths: Core Pure): Exam Questions

Show that

where  is a rational number to be found.

Using a substitution, that should be stated clearly, show that

where is an arbitrary constant and  and  are constants to be found.

Hence find, in exact form in terms of natural logarithms, the mean value of over the interval .

Assuming the derivative of prove that

Show that

where is an arbitrary constant and  and  are constants to be determined.

Hence find, in exact form, the mean value of over the interval

Show that

where is an arbitrary constant and  and  are constants to be found.

Hence show that the mean value of  over the interval is

Use the answer to part (b) to find the mean value, over the interval , of

where is a positive constant, giving your answer in the form where and  are constants and  is in terms of .