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

a) Explain why is an improper integral.

(b) Prove that

where and  are rational numbers to be determined.

(a)

Assuming the derivative of prove that

(b) Show that

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

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

Show that

where  is a rational number to be found

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

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

(b) Hence find, in exact form in terms of natural logarithms, the mean value of over the interval [0, 3].