1. integration

the integral from -1 to 2 of (cost +2tsint) dt

Thank you very much.

2. the integral from -1 to 2 of (cost +2tsint) dt

The indefinite integral first:
INT.[cost +2t*sint]dt
= sint +INT.[2t*sint]dt

Do the second integral by parts: INT.[u]dv = uv -INT.[v]du

Let u = 2t, so, du = 2dt
And dv = sint dt, so, v = -cost

= sint +2t*(-cost) -INT.[-cost](2dt)
= sint -2tcost +2sint +C
= 3sint -2cost +C -------------(i) <--------edit: should be -2tcost.

Then the definite integral:

INT.(-1 --> 2)[cost +2t*sint]dt
= [3sint -2cost] | (-1 --> 2)
= [3sin(2) -2cos(2)] -[3sin(-1) -2cos(-1)]
= [3sin(2) -2cos(2)] -[3sin(-1) -2cos(-1)]

-----------------------
Edit: Sorry, but I forgot about the t in -2tcost.

So let me correct the computations,

Then the definite integral:

INT.(-1 --> 2)[cost +2t*sint]dt
= [3sint -2tcost] | (-1 --> 2)
= [3sin(2) -2*2*cos(2)] -[3sin(-1) -2(-1)cos(-1)]
= [3sin(2) -4cos(2)] -[3sin(-1) +2cos(-1)]

integral(3sint-2cost) dt

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