What is the integral of sin(5x)cos(6x) ?

The integral of sin(5x)cos(6x) is 1/2 (-1/11 cos(11x)+cos(x))+C

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integral of sin(5x)cos(6x)

\int sin (5 x) cos (6 x) d x

\frac{1}{2} (- \frac{1}{11} cos (11 x) + cos (x)) + C

Solution steps

\int sin (5 x) cos (6 x) d x

Use the following identity:

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

= \int \frac{sin ( 5 x + 6 x ) + sin ( 5 x - 6 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 (5 x + 6 x) + sin (5 x - 6 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 (5 x + 6 x) d x + \int sin (5 x - 6 x) d x)

\int sin (5 x + 6 x) d x = - \frac{1}{11} cos (11 x)

\int sin (5 x - 6 x) d x = cos (x)

= \frac{1}{2} (- \frac{1}{11} cos (11 x) + cos (x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{11} cos (11 x) + cos (x)) + C

\int_{3}^{4} y^{3} + \frac{1}{4 y ^{3}} d y

\int \frac{9 x ^{8} + 6}{x ^{9} + 6 x} d x

f (x) = sin (7 ln (x))

\frac{d}{d x} (cos (3 cos (2 x)))

e^{- 2 s}

What is the integral of sin(5x)cos(6x) ?
The integral of sin(5x)cos(6x) is 1/2 (-1/11 cos(11x)+cos(x))+C