Get detailed and accurate responses to your questions with IDNLearn.com. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
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
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
