What is the integral of (ye^{xy}+2x-1) ?

The integral of (ye^{xy}+2x-1) is -y+2xy+e^{xy} 1/x y-e^{xy} 1/(x^2)+C

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integral of (ye^{xy}+2x-1)

\int (y e^{x y} + 2 x - 1) d y

- y + 2 x y + e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}} + C

Solution steps

\int y e^{x y} + 2 x - 1 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 1 d y + \int 2 x d y + \int e^{x y} y d y

\int 1 d y = y

\int 2 x d y = 2 x y

\int e^{x y} y d y = e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}}

= - y + 2 x y + e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}}

Add a constant to the solution

= - y + 2 x y + e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}} + C

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

\int_{1}^{\infty} \frac{x}{1 + x ^{2}} d x

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

t^{2} y^{^{'}} + t y = 3, y (1) = 4

\int_{0}^{1} x^{3} - x^{2} d x

What is the integral of (ye^{xy}+2x-1) ?
The integral of (ye^{xy}+2x-1) is -y+2xy+e^{xy} 1/x y-e^{xy} 1/(x^2)+C