## 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

1. this is ur required result in attachment

2. – 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]