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

The integral of (sin(nx))/n is -1/(n^2)cos(nx)+C

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integral of (sin(nx))/n

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

- \frac{1}{n ^{2}} cos (n x) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{n} \cdot \int sin (n x) d x

Apply u-substitution

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

= \frac{1}{n} \cdot \int \frac{sin ( u )}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \frac{1}{n} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{n} \cdot \frac{1}{n} (- cos (u))

Substitute back

u = n x

= \frac{1}{n} \cdot \frac{1}{n} (- cos (n x))

Simplify

\frac{1}{n} \cdot \frac{1}{n} (- cos (n x)) : - \frac{1}{n ^{2}} cos (n x)

= - \frac{1}{n ^{2}} cos (n x)

Add a constant to the solution

= - \frac{1}{n ^{2}} cos (n x) + C

y^{^{'}} = - \frac{1}{2} y

\int (\frac{12}{x} + \frac{x}{12}) d x

x y^{^{'}} + 3 y = \frac{2 e ^{4 x}}{x ^{2}}

\frac{d}{d x} (a x^{2} e^{- x})

\frac{d}{d x} (ln (\frac{3}{x ^{2} + 9}))

What is the integral of (sin(nx))/n ?
The integral of (sin(nx))/n is -1/(n^2)cos(nx)+C