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

f (x) = x^{\frac{6}{5}}

y^{^{''}} - y^{^{'}} + 2 y = 0

\frac{\partial}{\partial x} (e^{y} + x + x^{3} + 5)

\int_{- 1}^{1} (t^{3} - 9 t) d t

\frac{2}{x - 6}

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