What is the integral of sin(x-nx) ?

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

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integral of sin(x-nx)

\int sin (x - n x) d x

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

Solution steps

\int sin (x - n x) d x

Apply u-substitution

: \int \frac{sin ( u )}{1 - n} d u

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x - n x

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

Simplify

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

Add a constant to the solution

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

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