What is the integral of-2zsin(2z)+cos(2z) ?

The integral of-2zsin(2z)+cos(2z) is zcos(2z)+C

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integral of-2zsin(2z)+cos(2z)

\int - 2 z sin (2 z) + cos (2 z) d z

z cos (2 z) + C

Solution steps

\int - 2 z sin (2 z) + cos (2 z) d z

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 2 z sin (2 z) d z + \int cos (2 z) d z

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

\int cos (2 z) d z = \frac{1}{2} sin (2 z)

= - (- z cos (2 z) + \frac{1}{2} sin (2 z)) + \frac{1}{2} sin (2 z)

Simplify

- (- z cos (2 z) + \frac{1}{2} sin (2 z)) + \frac{1}{2} sin (2 z) : z cos (2 z)

= z cos (2 z)

Add a constant to the solution

= z cos (2 z) + C

What is the integral of-2zsin(2z)+cos(2z) ?
The integral of-2zsin(2z)+cos(2z) is zcos(2z)+C