What is the integral of (1+ye^{xy}) ?

The integral of (1+ye^{xy}) is x+e^{yx}+C

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integral of (1+ye^{xy})

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

x + e^{y x} + C

Solution steps

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

\int 1 d x = x

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

= x + e^{y x}

Add a constant to the solution

= x + e^{y x} + C

\int \frac{1}{x ( ln ( x ) ) ^{2}} d x

\frac{d}{d x} (\frac{e ^{x}}{x ^{2} + 1})

\int 18 x d x

\frac{d y}{d x}, 8 x^{3} + x^{2} y - x y^{3} = 9

\int \frac{1}{x ^{\frac{4}{5}} ( 1 + x ^{\frac{1}{5}} )} d x