What is the integral of-2ze^{2x}sin(yz)+6xy ?

The integral of-2ze^{2x}sin(yz)+6xy is 3xy^2+2e^{2x}cos(zy)+C

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integral of-2ze^{2x}sin(yz)+6xy

\int - 2 z e^{2 x} sin (yz) + 6 x y d y

3 x y^{2} + 2 e^{2 x} cos (zy) + C

Solution steps

\int - 2 z e^{2 x} sin (yz) + 6 x y d y

Apply the Sum Rule:

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

= \int 6 x y d y - \int 2 sin (zy) z e^{2 x} d y

\int 6 x y d y = 3 x y^{2}

\int 2 sin (zy) z e^{2 x} d y = - 2 e^{2 x} cos (zy)

= 3 x y^{2} - (- 2 e^{2 x} cos (zy))

Simplify

= 3 x y^{2} + 2 e^{2 x} cos (zy)

Add a constant to the solution

= 3 x y^{2} + 2 e^{2 x} cos (zy) + C

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What is the integral of-2ze^{2x}sin(yz)+6xy ?
The integral of-2ze^{2x}sin(yz)+6xy is 3xy^2+2e^{2x}cos(zy)+C