What is the integral of cos(nx)sin(nx) ?

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

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integral of cos(nx)sin(nx)

\int cos (n x) sin (n x) d x

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

Solution steps

\int cos (n x) sin (n x) d x

Apply u-substitution

: \int - \frac{u}{n} d u

= \int - \frac{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 \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - \frac{1}{n} \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (n x)

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

Simplify

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

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

Add a constant to the solution

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

\frac{15}{ln ( x )}

\int r^{4} d r

\frac{d}{d x} (2^{x^{2} - 1})

\frac{\partial}{\partial x} (\frac{F ( c , x , b ) \cdot cos ( c )}{x \cdot b})

\frac{d}{d x} (sin (2 cos (x)))

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