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

Question

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

in progress 0
2022-04-23T20:27:26+00:00 2 Answers 0

Answers ( 2 )

  1. Sadie
    0
    2022-04-23T20:28:56+00:00

    this is ur required result in attachment

  2. Ella
    0
    2022-04-23T20:29:01+00:00

    – cos t – u00b9/u2084cos 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]

    – – – – – – – – – –

    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

    1. About trigonometric identities https://brainly.com/question/1430645
    2. Using the product rule https://brainly.com/question/1578252
    3. The derivatives of the composite function https://brainly.com/question/6013189

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