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I'm having trouble solving this Riemann-Stieltjes integral:

$\int_{- \pi/4}^{\pi/4} f(x)dg(x)$ where $f(x):= \begin{cases} \frac{\sin^4x}{\cos^2x}{} &\text{if }x\ge0, \\{}\\ \frac1{\cos^3x} &\text{if }x<0.\end{cases}$

and $ g(x)=\begin{cases} \phantom{-} 1+\sin(x) &\text{if }-\pi/4 <x<\pi/4, \\ -1 &\text{otherwise}.\end{cases}$

I believe the only jump discontinuities are at $-\pi/4$ and $\pi/4$. Which $g=-1$ at both of those points. I'm struggling with the rest. What formula should I be using to compute the integral and what should my answer look like? Thanks for any help!

share|improve this question
Check this theorem. –  Mhenni Benghorbal May 3 '13 at 1:08
Can you elaborate how to solve the last integral in that theorem (related to my problem). I can see why I would use it since both f and g have no common discontinuities. –  Drake May 6 '13 at 13:51

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