Question

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

Answer 1

The solution of the function $\rm \int{sin(t) (1 + cos(t) )} \, dt$ 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;

$\rm \int{sin(t) (1 + cos(t) )} \, dt$

Multiply by sint in the function and simplify;

$\rm \int{sin(t) (1 + cos(t) )} \, dt\\\\\rm \int{sin(t) + sin(t)cos(t) \, dt$

Use trigonometric formulas for double angles:

$\rm 2sintcost =sin2t\\\\sin t cost =\dfrac{1}{2} sin2t$

Substitute the values in the function

$\rm \int{sin(t) (1 + cos(t) )} \, dt\\\\\rm \int{sin(t) + sin(t)cos(t) \, dt}\\\\ \int{sin(t) + \dfrac{1}{2} sin2t \, dt}$

And now we integrate this trigonometric form.

$\rm \int{sin(t) + \dfrac{1}{2} sin2t \, dt}\\\\ \int{sin(t) dt } +\dfrac{1}{2}\int{sin(2t)\, dt}\\\\-cost -\dfrac{1}{2} \times \dfrac{1 \times -cos2t}{2}\\\\-cost -\dfrac{{1 \times -cos2t}}{4}+c$

Hence, the solution of the given function is - cos t - ¹/₄cos 2t + c.

Learn more about indefinite integral here;

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