What is the integral of sin(3x)*cos(3x) ?

The integral of sin(3x)*cos(3x) is -1/6 cos^2(3x)+C

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integral of sin(3x)*cos(3x)

\int sin (3 x) \cdot cos (3 x) d x

- \frac{1}{6} cos^{2} (3 x) + C

Solution steps

\int sin (3 x) cos (3 x) d x

Apply u-substitution

: \int - \frac{u}{3} d u

= \int - \frac{u}{3} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{3} \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (3 x)

= - \frac{1}{3} \cdot \frac{cos ^{2} ( 3 x )}{2}

Simplify

- \frac{1}{3} \cdot \frac{cos ^{2} ( 3 x )}{2} : - \frac{1}{6} cos^{2} (3 x)

= - \frac{1}{6} cos^{2} (3 x)

Add a constant to the solution

= - \frac{1}{6} cos^{2} (3 x) + C

\int e^{y} - 1 d y

\frac{d}{d x} ((x - \frac{1}{x})^{\frac{5}{2}})

\frac{\partial}{\partial x} (4 sin (x y))

y = (\frac{2 x - 3}{( x ^{2} + 4 ) ^{2}})

(1, 4), (4, 1)

What is the integral of sin(3x)*cos(3x) ?
The integral of sin(3x)*cos(3x) is -1/6 cos^2(3x)+C