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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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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

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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...

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...

Answered by Eadaoin D. • Maths tutor

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