Some big numbers

Calculus Level 4

\[ \large \int \sin(101x) \sin^{99} x \, dx \]

If the indefinite integral above equals \[ \dfrac{\sin(ax) (\sin x)^b}{k} + C\] where \(a,b\) and \(k\) are constants and \(C\) is the arbitrary constant of integration, find \(a+b+k\).

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