Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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;
https://brainly.com/question/9829575
#SPJ4
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
