`g(x) = int_(1 - 2x)^(1 + 2x)(tsin(t))dt` Find the derivative of the function.

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Let's temporarily denote the antiderivative of `tsin(t)` as `A(t).`

Then the integral is equal to `A(1+2x)-A(1-2x)` and its derivative is


Recall what `A'` is and obtain


This can be simplified using `sin(a)+-sin(b)=2sin((a+-b)/2)*cos((a-+b)/2):`