CONTENTS

Solution: Reducing a Smooth Stieltjes Integral

Exercise: Reducing a Smooth Stieltjes Integral

Solution

Here g(x)=x3g(x)=x^3, so g′(x)=3x2g'(x)=3x^2. Therefore

∫02x d(x3)=∫023x3 dx=3[x44]02=12.\int_0^2 x\,d(x^3)=\int_0^2 3x^3\,dx=3\left[\frac{x^4}{4}\right]_0^2=12.

Takeaways

  • Smooth integrators reduce to weighted ordinary integrals.
  • The weight is g′(x)g'(x).
  • Stieltjes notation is a strict generalisation of Riemann notation.
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Solution - Reducing a Smooth Stieltjes Integral | q4quant.studio