Evaluate the indefinite integral. (use c for the constant of integration.) sin(t) 1 + cos(t) dt

The solution of the function [tex]\rm \int{sin(t) (1 + cos(t) )} \, dt[/tex] is - cos t - ¹/₄cos 2t + c.What is the indefinite integral?An indefinite integral is a function that practices the antiderivative of another function. It can be visually represented as an integral symbol, a function, and then a dx at the end.The given function is;[tex]\rm \int{sin(t) (1 + cos(t) )} \, dt[/tex]Multiply by sint in the function and simplify;[tex]\rm \int{sin(t) (1 + cos(t) )} \, dt\\\\\rm \int{sin(t) + sin(t)cos(t) \, dt[/tex]Use trigonometric formulas for double angles:[tex]\rm 2sintcost =sin2t\\\\sin t cost =\dfrac{1}{2} sin2t[/tex]Substitute the values in the function[tex]\rm \int{sin(t) (1 + cos(t) )} \, dt\\\\\rm \int{sin(t) + sin(t)cos(t) \, dt}\\\\ \int{sin(t) + \dfrac{1}{2} sin2t \, dt}\\\\[/tex]And now we integrate this trigonometric form.[tex]\rm \int{sin(t) + \dfrac{1}{2} sin2t \, dt}\\\\ \int{sin(t) dt } +\dfrac{1}{2}\int{sin(2t)\, dt}\\\\-cost -\dfrac{1}{2} \times \dfrac{1 \times -cos2t}{2}\\\\-cost -\dfrac{{1 \times -cos2t}}{4}+c[/tex]Hence, the solution of the given function is - cos t - ¹/₄cos 2t + c.Learn more about indefinite integral here;