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

Textbook Question

Chapter 5, 5.3 - Problem 56 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hello!

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

`A'(1+2x)*2-A'(1-2x)*(-2)=2*(A'(1+2x)+A'(1-2x)).`

Recall what `A'` is and obtain

`g'(x)=2*((1+2x)sin(1+2x)+(1-2x)sin(1-2x)).`

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

`g'(x)=2*(sin(1+2x)+sin(1-2x)+2x(sin(1+2x)-sin(1-2x)))=`

`=2*(2sin(1)cos(2x)+2x*2sin(2x)cos(1))=4(sin(1)cos(2x)+2xcos(1)sin(2x)).`

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