- cos t - ¹/₄cos 2t + c
Further explanation
We will evaluate the following integrals:
[tex]\boxed{ \ \int [sin(t)(1 + cos(t))] \ dt = ? \ }[/tex]
[tex]\boxed{ \ = \int [sin(t) + sin(t)cos(t)] \ dt \ }[/tex]
Use trigonometric formulas for double angles:
[tex]\boxed{ \ 2sin(t)cos(t) = sin2(t) \ } \rightarrow \boxed{ \ sin(t)cos(t) = \frac{1}{2}sin2(t) \ }[/tex]
[tex]\boxed{ \ = \int [sin(t) + \frac{1}{2}sin2(t)] \ dt \ }[/tex]
And now we integrate this trigonometric form.
[tex]\boxed{ \ = -cos(t) - \Big(\frac{1}{2}\Big) \Big(\frac{1}{2}\Big) \ cos \ 2(t) + c \ }[/tex]
Note that we use c for the constant of integration.
Thus the result is [tex]\boxed{ \ \int [sin(t)(1 + cos(t))] \ dt = -cos(t) - \frac{1}{4} \ cos \ 2(t) + c \ }[/tex]
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Notes
Please keep in mind the following basic trigonometric integrals:
- [tex]\boxed{ \ \int sin \ ax \ dx = - \frac{1}{a} \ cos \ ax + c \ }[/tex]
- [tex]\boxed{ \ \int cos \ ax \ dx = \frac{1}{a} \ sin \ ax + c \ }[/tex]
Learn more
- About trigonometric identities https://brainly.com/question/1430645
- Using the product rule https://brainly.com/question/1578252
- The derivatives of the composite function https://brainly.com/question/6013189
