What is the integral from y to y^3 of e^{x/y} ?

The integral from y to y^3 of e^{x/y} is y(e^{y^2}-e)

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integral from y to y^3 of e^{x/y}

\int_{y}^{y^{3}} e^{\frac{x}{y}} d x

y (e^{y^{2}} - e)

Solution steps

\int_{y}^{y^{3}} e^{\frac{x}{y}} d x

Apply u-substitution

: \int_{1}^{y^{2}} e^{u} y d u

= \int_{1}^{y^{2}} e^{u} y d u

Take the constant out:

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

= y \cdot \int_{1}^{y^{2}} e^{u} d u

Use the common integral:

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

= y [e^{u}]_{1}^{y^{2}}

Compute the boundaries:

e^{y^{2}} - e

= y (e^{y^{2}} - e)

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What is the integral from y to y^3 of e^{x/y} ?
The integral from y to y^3 of e^{x/y} is y(e^{y^2}-e)