What is the integral of cos(x+4y)-2xz ?

The integral of cos(x+4y)-2xz is -zx^2+sin(4y+x)+C

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integral of cos(x+4y)-2xz

\int cos (x + 4 y) - 2 x z d x

- z x^{2} + sin (4 y + x) + C

Solution steps

\int cos (x + 4 y) - 2 x z d x

Rewrite using trig identities

= \int cos (x) cos (4 y) - sin (x) sin (4 y) - 2 z 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

= \int cos (x) cos (4 y) d x - \int sin (x) sin (4 y) d x - \int 2 z x 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)

\int 2 z x d x = z x^{2}

= cos (4 y) sin (x) - (- sin (4 y) cos (x)) - z x^{2}

Simplify

cos (4 y) sin (x) - (- sin (4 y) cos (x)) - z x^{2} : - z x^{2} + sin (4 y + x)

= - z x^{2} + sin (4 y + x)

Add a constant to the solution

= - z x^{2} + sin (4 y + x) + C

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What is the integral of cos(x+4y)-2xz ?
The integral of cos(x+4y)-2xz is -zx^2+sin(4y+x)+C