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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-1 ln(x)+x n-1 =x n-1 =x n-1 (nln(x)+1) , for any positive real number x.Let ln(x)=t , thus x=e t and f(t)=e t(n-1) (nt+1) , fo...
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
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Answered by Fraser S. • Maths 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: A 2 + B 2 = C 2 (see diagram for explanation) Substitute the numbers given in the question into this equation:(4) 2 + (3) 2 = C 2 Expand out the ...
GW
Answered by Gina W. • Maths tutor
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how do ratios work

ratios are just fractions not written in fraction form, for example 8:9 ratio just means the first value is 8/(8+9) and the second value is 9/(8+9)if a question says that in a ratio of green apples to red ap...
MS
Answered by Minahil S. • Maths tutor
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Solve this equation: x²-x-6=0

This equation can be solved by factorisation. Start off with brackets : (x )(x ) =0. We need two numbers that multiply to give 6 and add/subtract to give 1. Start off with 1 and 6. They multiply to give 6 bu...
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Answered by Eadaoin D. • Maths tutor
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