What is the integral of e^{y/x} ?

The integral of e^{y/x} is xe^{y/x}+C

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integral of e^{y/x}

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

x e^{\frac{y}{x}} + C

Solution steps

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

Apply u-substitution

: \int e^{u} x d u

= \int e^{u} x d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= x \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= x e^{u}

Substitute back

u = \frac{y}{x}

= x e^{\frac{y}{x}}

Add a constant to the solution

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

x e^{- 3 x} sin (5 x)

x \to \infty lim (arctan (\frac{x + 1}{x + 2}))

\frac{\partial}{\partial x} (2 x^{2} + 3 y^{2})

\frac{d y}{d x} - 2 x y = 6 x

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