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

The integral of ye^{x/y} is y^2e^{x/y}+C

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integral of ye^{x/y}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int e^{u} y d u

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

Take the constant out:

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

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

Use the common integral:

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

= yy e^{u}

Substitute back

u = \frac{x}{y}

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

Simplify

yy e^{\frac{x}{y}} : y^{2} e^{\frac{x}{y}}

= y^{2} e^{\frac{x}{y}}

Add a constant to the solution

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

\int \frac{x}{12 - 9 x ^{2}} d x

\int_{1}^{4} x (2 + x) d x

w \to 0 lim (\frac{w ^{2}}{cos ( w ) - 1})

2 n d e^{2 e^{x}}

n = 1 \sum \infty \frac{n ^{2}}{3}