What is the integral of sin(12x)cos(5x) ?

The integral of sin(12x)cos(5x) is 1/2 (-1/17 cos(17x)-1/7 cos(7x))+C

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integral of sin(12x)cos(5x)

\int sin (12 x) cos (5 x) d x

\frac{1}{2} (- \frac{1}{17} cos (17 x) - \frac{1}{7} cos (7 x)) + C

Solution steps

\int sin (12 x) cos (5 x) d x

Use the following identity:

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

= \int \frac{sin ( 12 x + 5 x ) + sin ( 12 x - 5 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 + 5 x) + sin (12 x - 5 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 + 5 x) d x + \int sin (12 x - 5 x) d x)

\int sin (12 x + 5 x) d x = - \frac{1}{17} cos (17 x)

\int sin (12 x - 5 x) d x = - \frac{1}{7} cos (7 x)

= \frac{1}{2} (- \frac{1}{17} cos (17 x) - \frac{1}{7} cos (7 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{17} cos (17 x) - \frac{1}{7} cos (7 x)) + C

\int_{- \infty}^{0} \frac{1}{5 - 4 x} d x

\frac{d}{d x} (ln (13 x) + 13)

\frac{d y}{d x} = \frac{y ^{2} + 3 x x ^{2} + y ^{2}}{x y}

x \to 0 + lim (- csc^{2} (x))

\frac{d}{d x} (x^{2} (x + 3)^{10})

What is the integral of sin(12x)cos(5x) ?
The integral of sin(12x)cos(5x) is 1/2 (-1/17 cos(17x)-1/7 cos(7x))+C