What is the integral of sin(12x)cos(9x) ?

The integral of sin(12x)cos(9x) is 1/2 (-1/21 cos(21x)-1/3 cos(3x))+C

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integral of sin(12x)cos(9x)

\int sin (12 x) cos (9 x) d x

\frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{3} cos (3 x)) + C

Solution steps

\int sin (12 x) cos (9 x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int \frac{sin ( 12 x + 9 x ) + sin ( 12 x - 9 x )}{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 (12 x + 9 x) + sin (12 x - 9 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int sin (12 x + 9 x) d x + \int sin (12 x - 9 x) d x)

\int sin (12 x + 9 x) d x = - \frac{1}{21} cos (21 x)

\int sin (12 x - 9 x) d x = - \frac{1}{3} cos (3 x)

= \frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{3} cos (3 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{3} cos (3 x)) + C

x \to 0 + lim (arctan (\frac{1}{x}))

\int_{- 2}^{0} (u^{5} - u^{3} + u^{2}) d u

\int_{1}^{2} 2 r^{4} ln (r) d r

x \to 2 lim (csc (\frac{π x}{6}))

\frac{d ^{10}}{d x ^{10}} ((2 x + 5)^{7})

What is the integral of sin(12x)cos(9x) ?
The integral of sin(12x)cos(9x) is 1/2 (-1/21 cos(21x)-1/3 cos(3x))+C