What is the integral of sin((nt)/2) ?

The integral of sin((nt)/2) is -2/n cos((nt)/2)+C

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integral of sin((nt)/2)

\int sin (\frac{n t}{2}) d t

- \frac{2}{n} cos (\frac{n t}{2}) + C

Solution steps

\int sin (\frac{n t}{2}) d t

Apply u-substitution

= \int \frac{2 sin ( u )}{n} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{n} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= \frac{2}{n} (- cos (u))

Substitute back

u = \frac{n t}{2}

= \frac{2}{n} (- cos (\frac{n t}{2}))

Simplify

= - \frac{2}{n} cos (\frac{n t}{2})

Add a constant to the solution

= - \frac{2}{n} cos (\frac{n t}{2}) + C

What is the integral of sin((nt)/2) ?
The integral of sin((nt)/2) is -2/n cos((nt)/2)+C