What is the integral of 22cos(0.25x)sin(0.5x) ?

The integral of 22cos(0.25x)sin(0.5x) is 11(-1.33333…cos(0.75x)-4cos(0.25x))+C

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integral of 22cos(0.25x)sin(0.5x)

\int 22 cos (0.25 x) sin (0.5 x) d x

11 (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x)) + C

Solution steps

\int 22 cos (0.25 x) sin (0.5 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 22 \cdot \int cos (0.25 x) sin (0.5 x) d x

Use the following identity:

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

= 22 \cdot \int \frac{sin ( 0.5 x + 0.25 x ) + sin ( 0.5 x - 0.25 x )}{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 22 \cdot \frac{1}{2} \cdot \int sin (0.5 x + 0.25 x) + sin (0.5 x - 0.25 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

= 22 \cdot \frac{1}{2} (\int sin (0.5 x + 0.25 x) d x + \int sin (0.5 x - 0.25 x) d x)

\int sin (0.5 x + 0.25 x) d x = - 1.33333 \dots cos (0.75 x)

\int sin (0.5 x - 0.25 x) d x = - 4 cos (0.25 x)

= 22 \cdot \frac{1}{2} (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x))

Simplify

22 \cdot \frac{1}{2} (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x)) : 11 (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x))

= 11 (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x))

Add a constant to the solution

= 11 (- 1.33333 \dots cos (0.75 x) - 4 cos (0.25 x)) + C

\frac{d}{d x} (\frac{9 + 2 x}{7 - 4 x})

\int - \frac{sin ( x )}{x} d x

n = 1 \sum \infty (0.5)^{n}

\frac{d y}{d t} = 1 - y

What is the integral of 22cos(0.25x)sin(0.5x) ?
The integral of 22cos(0.25x)sin(0.5x) is 11(-1.33333…cos(0.75x)-4cos(0.25x))+C