What is the integral of sin(20x)cos(11x) ?

The integral of sin(20x)cos(11x) is 1/2 (-1/31 cos(31x)-1/9 cos(9x))+C

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integral of sin(20x)cos(11x)

\int sin (20 x) cos (11 x) d x

\frac{1}{2} (- \frac{1}{31} cos (31 x) - \frac{1}{9} cos (9 x)) + C

Solution steps

\int sin (20 x) cos (11 x) d x

Use the following identity:

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

= \int \frac{sin ( 20 x + 11 x ) + sin ( 20 x - 11 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 (20 x + 11 x) + sin (20 x - 11 x) d x

Simplify

= \frac{1}{2} \cdot \int sin (31 x) + sin (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 (31 x) d x + \int sin (9 x) d x)

\int sin (31 x) d x = - \frac{1}{31} cos (31 x)

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

= \frac{1}{2} (- \frac{1}{31} cos (31 x) - \frac{1}{9} cos (9 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{31} cos (31 x) - \frac{1}{9} cos (9 x)) + C

f^{^{'}} (x) = 3 x + 2

x \to 3 lim (\frac{1}{( x - 3 ) ^{3}})

\frac{d}{d t} (arctan (4 t))

\int cos (t) - 5 sin (t) d t

x \to 0 + lim (x ln (tan (5 x)))

What is the integral of sin(20x)cos(11x) ?
The integral of sin(20x)cos(11x) is 1/2 (-1/31 cos(31x)-1/9 cos(9x))+C