What is the integral of (sin(2x+y)-sin(x+y)) ?

The integral of (sin(2x+y)-sin(x+y)) is -1/2 cos(y+2x)+cos(y+x)+C

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integral of (sin(2x+y)-sin(x+y))

\int (sin (2 x + y) - sin (x + y)) d x

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

Solution steps

\int sin (2 x + y) - sin (x + y) 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

= \int sin (y + 2 x) d x - \int sin (y + x) d x

\int sin (y + 2 x) d x = - \frac{1}{2} cos (y + 2 x)

\int sin (y + x) d x = - cos (y + x)

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d s} (s^{2} y (s))

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

20 x - 50, y = 0, 2 x^{2}

\int (4 - r^{2}) r d r

f (x) = 9 x

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