What is the integral of ksin(kx)cos(kx) ?

The integral of ksin(kx)cos(kx) is -1/2 cos^2(kx)+C

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integral of ksin(kx)cos(kx)

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

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

Solution steps

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

Apply u-substitution

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

= k \cdot \int - \frac{u}{k} d u

Take the constant out:

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

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

Apply the Power Rule

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

= k (- \frac{1}{k} \cdot \frac{u ^{2}}{2})

Substitute back

u = cos (k x)

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of ksin(kx)cos(kx) ?
The integral of ksin(kx)cos(kx) is -1/2 cos^2(kx)+C