What is the integral of 2/n sin((nx)/2) ?

The integral of 2/n sin((nx)/2) is -4/(n^2)cos((nx)/2)+C

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integral of 2/n sin((nx)/2)

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

- \frac{4}{n ^{2}} cos (\frac{n x}{2}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = \frac{n x}{2}

= \frac{2}{n} \cdot \frac{2}{n} (- cos (\frac{n x}{2}))

Simplify

\frac{2}{n} \cdot \frac{2}{n} (- cos (\frac{n x}{2})) : - \frac{4}{n ^{2}} cos (\frac{n x}{2})

= - \frac{4}{n ^{2}} cos (\frac{n x}{2})

Add a constant to the solution

= - \frac{4}{n ^{2}} cos (\frac{n x}{2}) + C

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What is the integral of 2/n sin((nx)/2) ?
The integral of 2/n sin((nx)/2) is -4/(n^2)cos((nx)/2)+C