What is the integral of-ksin(nx) ?

The integral of-ksin(nx) is k 1/n cos(nx)+C

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integral of-ksin(nx)

\int - k sin (n x) d x

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

Solution steps

\int - k sin (n x) d x

Take the constant out:

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

= - k \cdot \int sin (n x) d x

Apply u-substitution

= - k \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

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

Use the common integral:

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

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

Substitute back

u = n x

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

Simplify

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

Add a constant to the solution

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