What is the integral of sin(4x-y) ?

The integral of sin(4x-y) is cos(-y+4x)+C

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integral of sin(4x-y)

\int sin (4 x - y) d y

cos (- y + 4 x) + C

Solution steps

\int sin (4 x - y) d y

Apply u-substitution

: \int - sin (u) d u

= \int - sin (u) d u

Take the constant out:

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

= - \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= - (- cos (u))

Substitute back

u = - y + 4 x

= - (- cos (- y + 4 x))

Simplify

= cos (- y + 4 x)

Add a constant to the solution

= cos (- y + 4 x) + C

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