What is the integral of 1+e^xy+xe^xy ?

The integral of 1+e^xy+xe^xy is ye^xx+x+C

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integral of 1+e^xy+xe^xy

\int 1 + e^{x} y + x e^{x} y d x

y e^{x} x + x + C

Solution steps

\int 1 + e^{x} y + x e^{x} 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 1 d x + \int y e^{x} d x + \int y e^{x} x d x

\int 1 d x = x

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

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

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

Simplify

x + y e^{x} + y (e^{x} x - e^{x}) : y e^{x} x + x

= y e^{x} x + x

Add a constant to the solution

= y e^{x} x + x + C

x \to 1 lim (3 x^{2} + 1)

\int (2 - y) d y

\frac{d ^{2}}{d x ^{2}} (6 ln (4 x))

x \to - 1 - lim (\frac{1}{x ^{2} + 1})

y = e^{x}, y = e^{5 x}, x = 1

What is the integral of 1+e^xy+xe^xy ?
The integral of 1+e^xy+xe^xy is ye^xx+x+C