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

The integral of sin(ax)cos(ax) is -1/(4a)cos(2ax)+C

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integral of sin(ax)cos(ax)

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

- \frac{1}{4 a} cos (2 a x) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{sin ( 2 x a )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (2 x a) d x

Apply u-substitution

: \int \frac{sin ( u )}{2 a} d u

= \frac{1}{2} \cdot \int \frac{sin ( u )}{2 a} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2 a} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{2 a} (- cos (u))

Substitute back

u = 2 x a

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

Simplify

\frac{1}{2} \cdot \frac{1}{2 a} (- cos (2 x a)) : - \frac{1}{4 a} cos (2 a x)

= - \frac{1}{4 a} cos (2 a x)

Add a constant to the solution

= - \frac{1}{4 a} cos (2 a x) + C

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What is the integral of sin(ax)cos(ax) ?
The integral of sin(ax)cos(ax) is -1/(4a)cos(2ax)+C