What is the integral of cos(x+4y) ?

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

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integral of cos(x+4y)

\int cos (x + 4 y) d x

sin (4 y + x) + C

Solution steps

\int cos (x + 4 y) d x

Rewrite using trig identities

= \int cos (x) cos (4 y) - sin (x) sin (4 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 cos (x) cos (4 y) d x - \int sin (x) sin (4 y) d x

\int cos (x) cos (4 y) d x = cos (4 y) sin (x)

\int sin (x) sin (4 y) d x = - sin (4 y) cos (x)

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

Simplify

cos (4 y) sin (x) - (- sin (4 y) cos (x)) : sin (4 y + x)

= sin (4 y + x)

Add a constant to the solution

= sin (4 y + x) + C

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