What is the integral of sin(14x)cos(7x) ?

The integral of sin(14x)cos(7x) is 1/2 (-1/21 cos(21x)-1/7 cos(7x))+C

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integral of sin(14x)cos(7x)

\int sin (14 x) cos (7 x) d x

\frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{7} cos (7 x)) + C

Solution steps

\int sin (14 x) cos (7 x) d x

Use the following identity:

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

= \int \frac{sin ( 14 x + 7 x ) + sin ( 14 x - 7 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 (14 x + 7 x) + sin (14 x - 7 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 (14 x + 7 x) d x + \int sin (14 x - 7 x) d x)

\int sin (14 x + 7 x) d x = - \frac{1}{21} cos (21 x)

\int sin (14 x - 7 x) d x = - \frac{1}{7} cos (7 x)

= \frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{7} cos (7 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{21} cos (21 x) - \frac{1}{7} cos (7 x)) + C

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\frac{\partial}{\partial x} (x^{2} + y)

\int (x^{2} - 2 x + 3)^{2} (x + 1) d x

\int x^{3} e^{2 x} d x

n = 1 \sum \infty \frac{n ^{5}}{2 ^{n}}

What is the integral of sin(14x)cos(7x) ?
The integral of sin(14x)cos(7x) is 1/2 (-1/21 cos(21x)-1/7 cos(7x))+C