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Let n be a positive integer. Find the continuous functions f:ℝ->ℝ with the property that integral from 1 to x of f(ln(t)) dt=x^n ln(x) for all positive real numbers x.

Differentiating the integral equation with respect to x we obtain: f(ln(x))=nx^n-1ln(x)+x^n-1=x

Answered by • Maths tutor

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integration by parts: x^-2lnx

u=lnx du/dx = 1/x dv/dx=x^-2 v= -1/x =uv - (integral of)vdu/dx (-lnx)/x + integral of x^-2 =(-lnx)/x - 1/x +c

Answered by Fraser S. • Maths tutor

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When completing creative writing questions, how do I get my readers attention?

When completing a creative writing piece, the aim is to draw your readers attention straight away so they are focused and interested from the beginning and therefore for the rest of your writing. This cou...

Answered by Kiera B. • English Language tutor

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If a right angled-triangle has sides A,B,C where A = 4 and B = 3 what is the value of side C?

This problem can be solved using Pythagoras Theorem equation: A² + B²= C² (see diagram for explanation)
Substitute the numbers given in the question into this equatio...

Answered by Gina W. • Maths tutor

2895 Views

How do I approach an unseen passage question?

Many A level Literature papers will feature unseen passage questions. These are extracts from a novel, or a play, in the genre you are studying, such as tragedy. I have 5 steps to make approaching these d...

Answered by Lauren R. • English Literature tutor

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