What is the integral of 2xe^y-ysin(x) ?

The integral of 2xe^y-ysin(x) is ycos(x)+e^yx^2+C

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integral of 2xe^y-ysin(x)

\int 2 x e^{y} - y sin (x) d x

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

Solution steps

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

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

\int 2 e^{y} x d x = e^{y} x^{2}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of 2xe^y-ysin(x) ?
The integral of 2xe^y-ysin(x) is ycos(x)+e^yx^2+C